Collective Modes and Thermodynamics of the Liquid State

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Collective modes and thermodynamics of the liquid state

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Rep. Prog. Phys. Rep. Prog. Phys. 79 (2016) 016502 (36pp) doi:10.1088/0034-4885/79/1/016502

RPPHAG

K Trachenko1 and V V Brazhkin2 016502 1 School of Physics and Astronomy, Queen Mary University of London, Mile End Road, London, E1 4NS, UK K Trachenko and V V Brazhkin 2 Institute for High Pressure Physics, RAS, 142190 Moscow, Russia

E-mail: [email protected] Collective modes and thermodynamics of the liquid state Received 14 March 2014, revised 25 September 2015 Accepted for publication 12 October 2015 Printed in the UK Published 22 December 2015

ROP Invited by Sean Washburn

Abstract 10.1088/0034-4885/79/1/016502 Strongly interacting, dynamically disordered and with no small parameter, liquids took a theoretical status between gases and solids with the historical tradition of hydrodynamic description as the starting point. We review different approaches to liquids as well as recent 0034-4885 experimental and theoretical work, and propose that liquids do not need classifying in terms of their proximity to gases and solids or any categorizing for that matter. Instead, they are 1 a unique system in their own class with a notably mixed dynamical state in contrast to pure dynamical states of solids and gases. We start with explaining how the first-principles approach to liquids is an intractable, exponentially complex problem of coupled non-linear oscillators with bifurcations. This is followed by a reduction of the problem based on liquid relaxation time τ representing non-perturbative treatment of strong interactions. On the basis of τ, solid-like high-frequency modes are predicted and we review related recent experiments. We demonstrate how the propagation of these modes can be derived by generalizing either hydrodynamic or elasticity equations. We comment on the historical trend to approach liquids using hydrodynamics and compare it to an alternative solid-like approach. We subsequently discuss how collective modes evolve with temperature and how this evolution affects liquid energy and heat capacity as well as other properties such as fast sound. Here, our emphasis is on understanding experimental data in real, rather than model, liquids. Highlighting the dominant role of solid-like high-frequency modes for liquid energy and heat capacity, we review a wide range of liquids: subcritical low-viscous liquids, supercritical state with two different dynamical and thermodynamic regimes separated by the Frenkel line, highly-viscous liquids in the glass transformation range and liquid-glass transition. We subsequently discuss the fairly recent area of liquid–liquid phase transitions, the area where the solid-like properties of liquids have become further apparent. We then discuss gas-like and solid-like approaches to quantum liquids and theoretical issues that are similar to the classical case. Finally, we summarize the emergent view of liquids as a unique system with a mixed dynamical state, and list several areas where interesting insights may appear and continue the extraordinary liquid story. Keywords: liquids, collective excitations, thermodynamics and dynamics of condensed matter (Some figures may appear in colour only in the online journal)

0034-4885/16/016502+36$33.00 1 © 2016 IOP Publishing Ltd Printed in the UK Rep. Prog. Phys. 79 (2016) 016502 Review Contents Harmonic theory...... 18 Comment on the phonon theory of liquid Introduction...... 2 thermodynamics...... 20 First-principles approach and its failure...... 4 Including anharmonicity and thermal expansion... 20 Relaxation time and phonon states in liquids: Including quantum effects...... 21 Frenkel’s reduction...... 6 Comparison with experimental data...... 22 Liquid relaxation time and phonon states...... 7 Phonon excitations at low temperature...... 22 Relationship to Maxwell relaxation theory...... 8 Heat capacity of supercritical fluids...... 23 Frenkel reduction...... 9 Frenkel line...... 23 Continuity of solid and liquid states Heat capacity above the Frenkel line...... 24 and liquid-glass transition...... 10 Heat capacity of liquids and system’s fundamental Hydrodynamic and solid-like elastic regimes length...... 25 of wave propagation...... 10 Evolution of collective modes in liquids: summary..... 26 Modifying elasticity: including hydrodynamics Viscous liquids...... 26 in elasticity equations...... 11 Energy and heat capacity...... 27 Modifying hydrodynamics: including elasticity Entropy...... 28 in hydrodynamic equations...... 12 Liquid-glass transition...... 29 Experimental evidence for high-frequency Phase transitions in liquids...... 30 collective modes in liquids...... 13 Quantum liquids: solid-like and gas-like Fast sound...... 14 approaches...... 31 Generalized hydrodynamics...... 16 Mixed and pure dynamical states: liquids, Comment on the hydrodynamic approach to liquids.....17 solids, gases...... 33 Phonon theory of liquid thermodynamics...... 18 Conclusions and outlook...... 34

Introduction systems with interesting, unique and rich dynamical and thermodynamic properties. In fact, understanding this richness Condensed matter physics as a term originated from adding liq- helps better understand the properties of gases and solids by uids to the then-existing field of solid state physics. Proposals delineating them as two limiting states of matter in terms of to do so precede what is often thought, and date back to the dynamics and thermodynamics. 1930s when J. Frenkel proposed to develop liquid theory as The long and extraordinary history of liquid research a generalization of solid state theory and unify the two states includes several notable discouraging assertions. One of the under the term ‘condensed bodies’ [1]. At the same time, the most important properties crucial to properly understanding seeming similarity of liquids and gases in terms of their ability liquids is that they are strongly-interacting systems. Particles to flow has led to the unified term‘ fluids’. Such a dual clas- in liquids are close enough to be within the reach of intera- sification of liquids is more than just semantics: it has given tomic forces as in solids, resulting in the condensed liquid rise to two fundamentally different ways of describing liquids state. The energy of a system with N particles and pair-wise theoretically in hydrodynamic and solid-like approaches. The interaction energy U(r) can be written as phase diagram of matter in figure 1 highlights the intermediate 3 Nρ location of liquids between solids and gases and hints at the E =+NkBT Ur()gr()dV (1) duality of their physical properties that will come out in our 22∫ detailed analysis. where ρ is number density and g(r) is pair distribution function. It is the intermediate state of liquids which has ultimately U(r) is strong and system-dependent; consequently, E or resulted in great difficulties when developing liquid theory other thermodynamic properties of the liquid are strongly sys- because well-developed theoretical tools for the two limiting tem-dependent. For this reason, Landau and Lifshitz assert [2] states of gases and solids failed. It is also the intermediate (twice, in paragraphs 66 and 74) that it is impossible to derive state of liquids and the combination of solid-like and gas-like any general equation describing liquid properties or their properties which continues to be remarkably intriguing for temperature dependence. Whatever approximation scheme theorists. According to figure 1, one can start in the gas state or method used, any approach aimed at deriving a generally above the critical point, move to the liquid state and end up in applicable result using equation (1), or evaluating the configu- the solid glass state (if crystallization is avoided) in a seem- rational part of the partition function, is destined to fail. ingly continuous way and without any qualitative changes of The above problem does not originate in strongly-interact- physical properties. This is a surprising observation from a ing solids because the smallness of atomic vibrations around theoretical point of view and signifies the intermediate state of the fixed reference lattice, crystalline or amorphous, ena- liquids and the duality of their physical properties. bles expansion of the potential energy in Taylor series. The At the end of this review, we will see that liquids need not harmonic term in this expansion, combined with the kinetic be thought of in terms of their proximity to solids or gases and term, gives the phonon energy of the solid consistent with do not require any other categorization: they are self-contained experimental heat capacities. These can be corrected by the

2 Rep. Prog. Phys. 79 (2016) 016502 Review

Figure 1. Phase diagram of matter with triple and critical points shown. Schematic illustration.

next-order terms in the Taylor series for potential energy. Figure 2. Experimental specific heat of liquid mercury ink B units [21, 22]. The x-axis is in the relative temperature units where Tm is Traditionally, this approach is deemed inapplicable to liquids the melting temperature. due to the absence of fixed reference points around which an expansion can be made. The problem also does not originate us that liquids not far from melting points are close to solids in weakly-interacting gases: they have no fixed reference in terms of density, bulk moduli, heat capacity and other main points but interactions are small so that the perturbation the- properties, but are very different from gases. ory is warranted. The focus of this review is on understanding liquid ther- Liquids have neither the small displacements of solids nor modynamic properties such as heat capacity and their rela- the small interactions of gases. Summarized aptly by Landau, tionship to collective modes. To be more specific and set the liquids have no small parameter. stage early, we show the experimental specific heat of liquid For this reason, we are seemingly compelled to treat liq- mercury in figure 2. We observe that cv starts from around uids as general strongly-interacting disordered systems, 3kB just above the melting point and decreases to about 2kB at where disorder is both static and dynamic, with no simpli- high temperature. As discussed below, this effect is very com- fying assumptions. In this spirit, large amount of work was mon and operates in over 20 different liquids we analyzed, aimed at elucidating the structure and dynamics of liquids. including metallic, noble, molecular and network liquids, and In comparison, the discussion of liquid thermodynamic prop- is present in complex liquids. The decrease of cv interestingly erties such as heat capacity is nearly non-existent. Indeed, contrasts the temperature dependence of cv in solids which is physics textbooks have very little, if anything, to say about either constant in the classical harmonic case or increases due liquid specific heat, including textbooks dedicated to liquids to anharmonicity or due to phonon excitations at low tempera- [2 12]. In an amusing story about his teaching experience in – ture. We also observe that liquid cv is significantly larger than the University of Illinois (UIUC), Granato recalls living in fear the gas value of 3 k in a wide temperature range in figure 2. about a potential student question about liquid heat capacity 2 B Notably, the commonly discussed Van Der Waals model of [13]. Observing that the question was never asked by a total of liquids gives c = 3 k [2], the ideal gas value. The same result 10 000 students, Granato proposes that ‘...an important defi- v 2 B ciency in our standard teaching method is a failure to mention holds for another commonly discussed model of liquids, hard sufficiently the unsolved problems in physics. Indeed, there spheres, as well as for several other more elaborate models. is nothing said about liquids (heat capacity) in the standard Clearly, real liquids have an important mechanism at opera- introductory textbooks, and little or nothing in advanced texts tion that significantly affects their cv and that is missed by as well. In fact, there is little general awareness even of what several common liquid models. the basic experimental facts to be explained are’. It is probably Notwithstanding the theoretical difficulties involved in fair to say that the question of liquid heat capacity would be treating liquids, we rely on the known result that low-energy out of the comfort zone not only for general condensed matter states of a strongly-interacting system are collective excita- practitioners but also for many working in the area related to tions or modes (throughout this review, we use terms ‘pho- the liquid state such as soft matter. nons’, ‘modes’ and ‘collective excitations’ interchangeably Historically, thermodynamic properties of liquids have depending on context and common usage). In solids, collec- been approached from the gas state, a seemingly appropri- tive modes, phonons, play a central role in the theory, includ- ate approach in view of liquid fluidity. For example, common ing the theory of thermodynamic properties. Can collective approaches start with the kinetic energy of the gas and aim modes in liquids play the same role? It is from this perspective to calculate the potential energy using the perturbation the- that we review collective modes in liquids. In our review, we ory. The dynamical properties of liquids are discussed on the emphasize the main different approaches to collective modes basis of hydrodynamic theory where the elements of solid-like in liquids and list starting equations in each approach. We do behaviour are introduced as a subsequent correction [4–12, not discuss details of how the field has branched out over time; 14]. This is in interesting contrast to experiments informing that formidable task is outside the scope of this paper. To a

3 Rep. Prog. Phys. 79 (2016) 016502 Review large extent, this was done in earlier textbooks and reviews in notably mixed dynamical state. Therefore, the emergent [4–12, 14]. picture of liquids is that they do not need classifying on the We focus on real rather than model liquids, measurable basis of their proximity to fluid gases or solids, or any other effects and take a pragmatic approach to understand the main compartmentalizing for that matter. Instead, they should be experimental properties of liquids such as heat capacity and considered as distinct systems in the mixed state of particle provide relationships between different physical properties. dynamics, the state that should serve as a starting point for Throughout this review, we seek to make connections between liquid description. Moreover, we will see that appreciating the different areas of physics that help understand the problem. mixed state of particle dynamics in liquids helps understand We are not trying to be completely comprehensive, focusing gases and solids better as two limiting and dynamically pure instead on providing a pedagogical introduction, interpreting states. This point is particularly useful for understanding the previous basic results and fundamental equations and explain- supercritical matter. ing recent advances. Our discussion includes original work We conclude with possible future work which may bring not reported previously as well as results from our published new understanding and advance the remarkable liquid story. work. Before we start, we comment on several terms used in As already mentioned, the long and extraordinary his- this review. Traditionally, the term ‘liquids’ is used for tory of liquid research is related to problems of theoretical subcritical conditions on the phase diagram. The systems description. The fundamental problem of the first-principles above the critical point are often referred to as ‘supercritical description of liquids is not generally discussed, so we start fluids’. We continue to use these terms in our review where with explaining that this problem is due to the intractabil- we also propose that the supercritical system in fact consists ity of the exponential complexity of finding bifurcations and of two states in terms of particle dynamics and physical prop- stationary points in the system of coupled non-linear oscilla- erties: a ‘rigid liquid-like’ state below the Frenkel line and tors. We then discuss how the problem can be reduced using a ‘non-rigid gas-like fluid’ state above the line. We use the Frenkel’s idea of liquid relaxation time. On this basis, several term ‘glass’ to commonly denote a very viscous liquid which important assertions can be made regarding the continuity stops flowing at the typical experimental time scale. The term of liquid and solid states and the propagation of solid-like ‘viscous liquid’ commonly refers to liquids in the glass trans- collective modes in liquids. We subsequently review how formation range, implying viscosity considerably higher than collective modes can be studied by either incorporating elastic that in, for example, water at ambient conditions. The term is effects in hydrodynamic equations or viscous effects in elas- quantitatively defined at the beginning of the section ‘viscous ticity equations. We find the same results in both approaches, liquids’. supporting the view that the historical hydrodynamic description of liquids is not unique and that a solid-like description is equally justified. This assertion becomes more specific when First-principles approach and its failure we review and comment on generalized hydrodynamics. As far as liquid thermodynamics is concerned, it turns out that The absence of a small parameter in liquids pointed out in the the solid-like elastic regime is the relevant one because high- Introduction, is one perceived reason that makes the theoreti- frequency solid-like collective modes contribute most to the cal description of liquids difficult. It tells us why perturbation- energy. based approaches that are successful in solids and gases do We then proceed to review recent experimental evidence not work in liquids. Yet it is interesting to explore the actual for high-frequency solid-like collective modes in liquids and reason for the difficulty of constructing a first-principles the- discuss their similarity to those in solids. ory of liquids using the same microscopic approach as in the We subsequently discuss how the evolution of collective solid theory. As far as we know, this point is not discussed in modes in liquids can be related to liquid energy and heat capac- textbooks [1–11, 12]. ity in widely different liquid regimes: low-viscous subcritical Below we show that the challenge for the first-principles liquids; high-temperature supercritical gas-like fluids; highly- description of liquids can be well formulated in the language viscous liquids in the glass transformation range; and systems of non-linear theory where it acquires a specific meaning. In at the liquid-glass transition. In all cases, high-frequency this language, the challenge is related to the intractability of modes govern the main thermodynamic properties of liquids the exponentially complex problem involved in solving a large such as energy and heat capacity and affect other interesting number of coupled non-linear equations. effects such as fast sound. First-principles treatment of collective modes in a solid is The solid-like properties of liquids have additionally based on solving coupled Newton equations of motion for N become apparent in the recently accumulated and reviewed atoms. We assume that the atoms oscillate around fixed lattice data on liquid–liquid phase transitions. We finally discuss the points qi0, and introduce atomic coordinates qi and displace- gas-like and solid-like approach in quantum liquids and inter- ments xqi =−iiq 0. The potential energy is expanded in series esting issues regarding the operation of Bose–Einstein con- as far as quadratic terms: densates in real liquids. 1 At the end of this review we will see that most important U =+Uk0 ∑ ij xxjk (2) 2 properties of liquids and supercritical fluids can be consist- ij ently understood in the picture in which these systems are Writing the equations of motion as

4 Rep. Prog. Phys. 79 (2016) 016502 Review is multi-well, however the minima and energy profiles can be assumed to be close to their averages in a homogeneous system so that the double-well potential in figure 3 suffices. To model the double-well energy, the harmonic expansion (2) needs to be extended to include higher terms, at which point the equations of motion become non-linear. The simpler form often considered includes the third and fourth powers of 34 xi, ‘−xx+ ’: 1 UU=+0 ∑ kxij jkx 2 ij ++∑∑kxijl ijxxl kxijl ijxxlmx + ... (6) ijl ijlm or, if a symmetric form of U is preferred, the higher-order 46 potential can be written in ‘−xx+ ’ or similarly symmetric form as in figure 3. Figure 3. Double-well potential describing the particle motion in At small enough energy or temperature of particles motion, liquids and involving jumps between different quasi-equilibrium equation (6) is used to describe the effects of anharmonicity positions. of atomic motion in solids using the perturbation theory. The main results include the correction to the Dulong-Petit result ∑ mxii¨ +=kxji i 0 (3) of solids, thermal expansion and modification of the phonon i spectrum, phonon scattering and so on. Unfortunately, the and seeking the solutions as xbkk= expi()ωt gives the charac- quantitative evaluation of anharmonicity effects has remained teristic equation for the eigenfrequencies a challenge, with the frequent result that the accuracy of leading-order anharmonic perturbation theory is unknown km2 0 ij −=ω i (4) and the magnitude of anharmonic terms is challenging to Equation (4) gives most detailed information about collec- justify [23–27]. Experimental data such as phonon lifetimes tive modes in the system, and returns 3N eigenfrequencies, and frequency shifts can provide quantitative estimates for ranging from the lowest frequency set by the system size to anharmonicity effects and expansion coefficients in particular, the largest frequency in the system, often referred to as Debye although this involves complications and limits the predictive frequency (note that Debye frequency is the result of quad- power of the theory [23]. ratic approximation to the energy spectrum, and is somewhat The real problem is at higher energy where the anharmo- lower than the maximal frequency of the real spectrum). Each nicity in equation (6) is not small and jumps between different atomic coordinate can be expressed as a superposition of nor- minima in figure 3 become operative, as they do in liquids. mal coordinates as Here, the perturbation approach does not apply, and we enter the realm of non-linear physics [28, 29]. The illustrative exam- xkk=∆∑ ααΘ (5) ple is the simplest system of two coupled Duffing oscillators α with the energy (see, e.g. [28]): where Θ = Re Ctexpi are normal coordinates, C are αα((ωα )) α 2 2 arbitrary complex constants and ∆ are minors of the deter- ⎛ 1 22αβ4⎞ ε 2 kα E =+∑⎜⎟xx˙ii−+xxi ()12− x (7) minant (4) [15]. This result is central for the development of i=1⎝ 2 24⎠ 2 many areas in the solid state theory. and the equations of motion Note that the above treatment does not assume a crystalline 2 3 lattice. Crystallinity, if present, is the next step in the treat- xx¨11++αβε ()xx12−−x1 = 0 (8) ment enabling to write the solution as a set of plane waves 2 3 xx¨22++αβε ()xx21−−x2 = 0 with xk∝ expi()an , where k is the wavenumber and a is the shortest interatomic separation, and derive dispersion curves where ε is the coupling strength. for model systems. This model is not integrable, and can not be solved analyti- To continue to use the first-principles description of liquids, cally but using approximations only. However, a simpler model 1 dxn we need to account for particle rearrangements in liquids. As can written in terms of variables ψnn=+ω0x i , discussed in the next section, particle dynamics in the liquid 2ω0 ()dt where ω is the frequency of the uncoupled oscillator: consists of small solid-like oscillations around quasi-equi- 0 librium positions and diffusive jumps to new neighbouring dψ1 Ω 2 i =+ωψ01 ()ψψ12−−αψ1 ψ1 locations. This corresponds to potential energy of the double- d2t well form shown in figure 3 which endows particles with both dψ2 Ω 2 i =+ωψ02 ()ψψ21−−αψ2 ψ2 (9) oscillatory motion and thermally-induced jumps between dif- d2t ferent minima. Note that in an equilibrium liquid, each diffus- 2 ing particle visits many minima, hence the potential energy where the last terms represent the non-linearity and Ω= ε . ω0 5 Rep. Prog. Phys. 79 (2016) 016502 Review simple non-linear system, the liquid-like motion emerges as a bifurcation of the solid-like solution. The DST model (9) is not identical to the original simple system of coupled Duffing oscillators (8). The difference with the DST model is that, due to the non-integrability of (8), islands of chaotic dynamics appear on the phase map and grow with the system energy. The excitations in the original model (8) can only be found using approximate techniques. However, the DST model is close to (8) for small oscillation amplitudes and small couplings ε. This proximity between the two models is used to assert the same qualitative result, the emergence of the bifurcation of solutions. We note the result from this discussion to which we return below: the bifurcation in the original model (8) emerges at energies E ∝ ε 2 or amplitudes x ∝ ε , the result which is not unexpected: the energy of coupling needs to be surmounted in order to break away from the low-energy solid-like solution. The real problem appears when the number of non-linear oscillators, N, increases. The analysis of N = 3 non-linear coupled oscillators is complicated from the outset by the fact that the corresponding DST model is non-integrable to begin Figure 4. The phase portrait of the discrete trapping model (9) at with. The approximations involved in the increasingly com- two different energies. Independent variables u and v are functions plicated analysis of stationary states, new bifurcations emerg- of ψ1 and ψ2 in equation (9), and describe the trajectories of two ing from these states and corresponding collective modes coupled non-linear oscillators. At low energy (top), two stationary become harder to control. The results from computer mode- points ‘(a)’ and ‘(b)’ remain unchanged as in the case of two coupled linear oscillators, corresponding to weak non-linearity. ling indicate the emergence of many unanticipated modes and At high energy (bottom), the bifurcation takes place: point ‘(a)’ chaotic behaviour at higher energy. The problem significantly becomes an unstable saddle point and two new stationary points increases for N = 4, including finding new stationary points ‘(c)’ appear. Schematic illustration, adapted from [28]. and related collective modes, analyzing non-trivial branching of next-generation bifurcations and so on. For larger N, only Equations (9) is known as the discrete self-trapping (DST) approximate qualitative observations can be made regarding model, and is one of the rare examples in non-linear physics the energy spectrum, energy localization and emerging collec- that are exactly solvable analytically. The important results tive modes. This is done on the basis of approximations and can be summarized as follows. At low energy, the stationary insights from N = 2–4 [28]. points on the map (xx, ˙) (or on the map of two other inde- Importantly, the number of stationary states and bifurca- pendent dynamical variables) do not change, and the motion tions exponentially increases with N. The problem of finding remains oscillatory and similar to the linear case. The char- stationary states, bifurcations, collective modes and their evo- acter of oscillations qualitatively changes at a certain energy lution with the system’s energy is exponentially complex and that depends on Ω: the old stationary point becomes an unsta- intractable for arbitrary N [28]. α ble saddle point, and a new pair of stable stationary points Therefore, the failure of the first-principles treatment of emerge, separated by the energy barrier [28]. This corresponds liquids at the same level as equation (3) for solids has its ori- to the bifurcation point, the emergence of new solutions as a gin in the intractability of the exponentially complex problem result of changes of parameters in the dynamical system. This of calculating bifurcations, stationary points and collective is illustrated in figure 4. modes in a large system of coupled non-linear equations. An accompanying interesting insight is that contrary to the linear harmonic case, the energy is not equally partitioned Relaxation time and phonon states in liquids: between the oscillating points but can localize at one point, Frenkel’s reduction reflecting the more general insight that the superposition prin- ciple no longer works in non-linear systems in general. It is fitting to discuss terms such as‘ collective modes’, ‘pho- The importance of the above result is that it demonstrates nons’ and other quasi-particles in relation to Frenkel’s work that the first-principles treatment of the non-linear equa- because he was involved in coining and disseminating these tions of motion gives rise, via the bifurcation at high energy, terms. For example, the term ‘phonon’, attributed to Tamm, a new qualitatively different solution: instead of oscillating first appeared in print in Frenkel’s 1932 publication [30]. around a fixed position at low energy as in a solid, a particle Frenkel’s ideas occupy a significant part of our discussion. starts to move between two stable stationary points at high This might appear unusual to the reader, in view that this is energy, corresponding to the liquid-like motion of particles not the case in other liquid textbooks [2–12]. Frenkel’s work between two minima in figure 3. It proves that in the most is not unknown but why would we want to delve into it in

6 Rep. Prog. Phys. 79 (2016) 016502 Review

modern generation of scientists... He asks: ‘what is really hap- pening and how can this be explained?’ ’

Liquid relaxation time and phonon states Throughout this paper, we are using terms such as collective modes and phonons inter-changeably. Their meaning will be Figure 5. Illustration of a particle jump between two quasi- equilibrium positions in a liquid. These jumps take place with a clarified in the later sections where we will also comment on period of τ on average. the issue of dissipation of harmonic waves in disordered systems including glasses and liquids. detail now? We find that many discussions of liquids either do Dating to 1926 [32] and developed in his later book [1], not mention Frenkel’s work (see, e.g. [16–20]) or mention it main ideas of Frenkel on liquids preceded the advance of in an irrelevant context, yet they develop many ideas which, the non-linear theory discussed earlier. Frenkel’s discussion when stripped of details, are essentially due to Frenkel to a includes many important ideas, of which we review only large extent. This will become apparent in this review. More those relevant to understanding collective modes and different importantly, we find that, combined with recent experimental regimes of wave propagation in liquids. evidence, Frenkel’s work related to collective modes in liq- Frenkel was naturally led to liquid dynamics by his work uids gives a constructive tool to develop a predictive thermo- on defect migration in solids, and viewed the two processes dynamic theory of liquids. as sharing important qualitative properties. The migration rate Frenkel proposed a number of new ideas of how to under- of defects in a solid (crystalline or amorphous) is governed stand liquids emphasizing their ‘gas-like’ and ‘solid-like’ the potential energy barrier U set by the surrounding atoms. properties [1]. Some of the ideas such as the ‘hole theory’ At fixed volume of the ‘cage’ formed by the nearest neigh- of liquids were not followed or developed, perhaps for the bours, U is very large for the diffusion event to occur in any reason that the picture was qualitative and without links to reasonable time. However, the cage thermally oscillates and experimental data. It should be noted that the experimental periodically opens up fast local diffusion pathways. If ∆r is data on liquids at the time was only very basic so Frenkel’s the increase of the cage radius required for the atom to jump theoretical work was truly pioneering. However, other ideas from its case (see figure 5), U is [1]: discussed in Frenkel book and his earlier papers on liquids 2 (10) transformed the field in a way which is not fully appreciated U =∆8πGr r even today. where r is the cage radius and G is shear modulus. Note that This transformation proceeded slowly and sporadi- when a sphere expands in a static elastic medium, no com- cally over the last 80–90 years since Frenkel’s work, during pression takes place at any point. Instead, the system expands which alternative approaches to liquids were developing and by the amount equal to the increase of the sphere volume [1], Frenkel’s ideas forgotten and surfaced anew more than once resulting in a pure shear deformation. The strain components (see, e.g. [16, 18, 20]). In our view, Frenkel was too ahead u from an expanding sphere (noting that u → 0 as r → ∞) are 3 3 of his time. A transformative idea, proposed and experimen- ubrr =−2/r , uuθθ ==φφ br/ [33], giving pure shear uii = 0. tally confirmed within a generation of scientists has a larger As a result, the energy to statically expand the sphere depends chance of succeeding as compared to the Frenkel’s case on shear modulus G only. where the new idea was proposed long before its confirma- Frenkel considered the above picture applicable to liquids tion. For example, his proposal that liquids are able to support as well as solids, and introduced liquid relaxation time τ as solid-like longitudinal and transverse modes with frequencies the average time between particle jumps at one point in space extending to the highest Debye frequency implies that liquids in a liquid. are just like solids (solid glasses) in terms of their ability to The range of τ is bound by two important values. If crys- sustain collective modes. Therefore, main liquid properties tallization is avoided, τ increases at low temperature until such as energy and heat capacity can be described using the it reaches the value at which the liquid stops flowing at the 2 3 same first-principles approach based on collective modes as experimental time scale, corresponding to τ = 10 –10 s solids—an assertion that is considered very unusual. The evi- and the liquid-glass transition [34, 35]. At high temperature, dence for this has come only recently because liquids turned τ approaches its minimal value given by Debye vibration out to be too hard to probe experimentally. The evidence has period, τD ≈ 0.1 ps, when the time between the jumps becomes started to mount only after powerful x-ray synchrotrons were comparable to the shortest vibrational period. Frenkel’s pic- deployed, and more than 80 years after Frenkel’s first pub- ture has been confirmed in numerous molecular dynamics lished paper on the subject. simulations of liquids which, since early days of computer Frenkel’s work on liquids is interestingly described by Sir modeling [36], observed and studied particle jumps and tran- Mott [31]: sitions between different minima. The operation of particle ‘Frenkel was a theoretical physicist. By this I am stressing jumps in liquids is often referred to as ‘relaxation process’. that he was primarily and most of all interested in what is hap- With a remarkable physical insight, Frenkel proposed the pening in real systems, and the mathematics he used served following simple picture of vibrational states in the liquid. At his physics and not otherwise as is sometimes the case for the times significantly shorter thanτ , no particle rearrangements

7 Rep. Prog. Phys. 79 (2016) 016502 Review take place. Hence, the system is a solid glass describable by is the sum of viscous and elastic strains [38]. The y-gradient of

equations (3) and (4) and supports one longitudinal mode ∂vx Pxy horizontal velocity vx due to viscous deformation is = , and two transverse modes. At times longer than τ, the system ∂y η is a flowing liquid, and hence does not support shear stress where Pxy is shear stress and η is viscosity. The gradient of dP or shear modes but one longitudinal mode only as any elas- velocity due to elastic deformation is ∂vx = 1 xy where G is ∂yGdt tic medium (in a dense liquid, the wavelength of this mode shear modulus. When both viscous and elastic deformations extends to the shortest wavelength comparable to interatomic are present, the velocity of a layer vx is the sum of the two separations as discussed below). This is equivalent to assert- velocities, giving: ing that the only difference between a liquid and a solid glass is that the liquid does not support all transverse modes as the ∂vx 1 dPxy Pxy =+ (12) solid, but only those above the Frenkel frequency ωF: ∂yGdt η 1 The presence of both viscous and elastic response has been ω >=ωF (11) τ subsequently called ‘viscoelastic’ response, and is commonly used at present. where we omit the factor of 2π in ω = 2π for brevity and for τ When external perturbation stops and vx = 0, equation (12) the reason that in liquids the range of τ spans 16 orders of gives magnitude, making a small constant factor unimportant. Equation (11) implies that liquids have solid-like ability to ⎛ t ⎞ PPxy =−0 exp⎜ ⎟ support shear stress, with the only difference that this ability ⎝ τM ⎠ exists not at zero frequency as in solids but at frequency larger η τM = (13) than ωF (below we often use the term ‘solid-like’ to denote the G property in (11)). This was an unexpected insight at the time, where τ is Maxwell relaxation time. and took many decades to prove experimentally as discussed M Frenkel has proposed that the time constant in equa- below. It also posed a fundamental question about the differ- tion (13), τ , is related to liquid relaxation time τ he introduced ence between solids and liquids: liquids are different from M (the time between particle rearrangements), and concluded solids by the value of ω only which is a quantitative dif- F that ττ≈ . Then, relaxation of shear stress in a viscoelastic ference rather than a qualitative one. In Frenkel s view, this M ’ liquid is exponential with Frenkel s liquid relaxation time τ: reflected the continuity of liquid and solid states, the question ’ that is still debated in the context of the problem of liquid- ⎛ t ⎞ PPxy =−0 exp⎜⎟ (14) glass transition. We will discuss this in the next sections. ⎝ τ ⎠ The longitudinal mode remains propagating in the Frenkel s ’ The second equation in (13) where τ is used instead of τ picture based on : density fluctuations exist in any interacting M τ is often called the Maxwell relationship: system. We will see below that in real dense liquids, experiments have ascertained that the longitudinal vibrations extend η = G∞τ (15) to the largest Debye frequency as in solids. However, the Here, G∞ is the ‘instantaneous’ shear modulus. G∞ is presence of relaxation process and τ differently affects the understood to be the shear modulus at high frequency at which propagation of the longitudinal collective modes in different the liquid supports shear stress. In practice, this frequency can 1 1 regimes ω > and ω < , as discussed in the next sections. be taken as the maximal frequency of shear waves present in τ τ We note that the separation of particle motion in the liquid the liquid, comparable to Debye frequency [39]. into oscillatory and diffusive jump motion works well for The activation energy for particle jumps in the liquid can liquids with large τ (or viscosity, see next section). For smaller be calculated using equation (10), but with the proviso that G τ at high temperature, jumps can become less pronounced and is the shear modulus at high-frequency. oscillations increasingly anharmonic. The disappearance of Experimentally, shear stress and various correlations in oscillatory component of particle motion can be related to the viscous liquids (liquids where ττ D; see section ‘viscous Frenkel line discussed in the later section. liquids’ for more detailed discussion below) and glasses decay We also note that the concept of τ implies average relaxa- according to the stretched-exponential relaxation (SER) law tion time. In real liquids, there is a distribution of relaxation rather than pure exponential as in equation (14): times as is widely established in experiments such as dielec- β tric spectroscopy (see, e.g. [37]). ⎛ ⎛ t ⎞ ⎞ f ∝−exp⎜ ⎜⎟⎟ (16) ⎝ ⎝ τ ⎠ ⎠ where f is a decaying function such as P in (14) and is the Relationship to Maxwell relaxation theory xy β stretching parameter conforming to 0 <<β 1. Here, we discuss the important relationship between the anal- First observed by Kohlrausch around the time of develop- ysis of Frenkel and Maxwell. Maxwell proposed that a body is ment of Maxwell relaxation theory [40], the physical origin generally capable of both elastic and viscous deformation and, of SER has been widely discussed [34, 41, 42]. It is believed under external perturbation such as shear stress, the total strain that SER is as a result of cooperativity of molecular relaxation

8 Rep. Prog. Phys. 79 (2016) 016502 Review emerging in the viscous regime. Here, ‘cooperativity’ is not well-defined but can be identified with the elastic interaction between particle rearrangement events via high-frequency waves they induce [43, 44]. Regardless of whether the relaxation is exponential or stretched-exponential, the decay of shear stress and other correlation function takes place with a characteristic time τ in both (14) and (16).

Frenkel reduction

It is interesting to discuss the meaning of Frenkel’s theory from the point of view of a first-principles description of liquids. This theory is not a first-principles description at level (3) and (4) but, as discussed in the earlier section, the first- principles treatment of liquid collective modes is exponentially complex and not tractable. Instead, this approach singles Figure 6. Relaxation time of salol measured in dielectric relaxation out the main physical property of liquids (τ, or viscosity, see experiments [56]. equation (15)) which governs the relative contributions of oscillatory and diffusive motion and which ultimately controls In our discussion of generalized hydrodynamics below, the phonon states in the liquid. This reduces the exponentially we will see that the introduction of relaxation process and large problem to one physically relevant parameter. We call it solid-like features in the hydrodynamic equations is done at the ‘Frenkel reduction’ [45]. the same level as in equation (14) in the Frenkel theory, by Implicit in this reduction is a physically reasonable assuming the exponential decay of different correlation func- assumption that quasi-equilibrium states and the local particle tions with the decay time τ. surroundings of jumping atoms in a homogeneous liquid are We emphasize that τ is readily measured using several equivalent, and that fluctuations in a statistically large system well-established experiments including dielectric relaxation can be ignored [2]. In the language of non-linear theory, the experiments, NMR, positron annihilation spectroscopy and so reduction lies in assuming that emerging new bifurcations on. τ can also be derived from viscosity measurements using and stationary states at all generations produce physically equation (15) using widely available techniques including the equivalent states on average. This implies that as temperature classic Stokes experiments applicable to many types of liquids (or energy) increases, the conditions governing particle jumps including at high pressure and temperature [47]. τ can also be can be considered approximately the same everywhere in the calculated in molecular dynamics simulations as, for exam- system. Therefore, particle dynamics is governed by the tem- ple, time decay of various correlation functions. In figure 6 we perature-activated jumps as the dynamics of point defects in show τ measured in salol over many orders of magnitude as an solids: example, and comment on it in the next section. In essence, Frenkel reduction introduces a cutoff frequency ⎛U ⎞ ττ= D exp⎜⎟ (17) ωF (see (11)) above which the liquid can be described by the ⎝ T ⎠ same first-principles equations of motion as the solid in equa- where U is given by (10). tions (3) and (4). Therefore, liquid collective modes include It is generally agreed that τ and viscosity in liquids are both longitudinal and transverse modes with frequency above indeed governed by the temperature-activated process, with a ωF in the solid-like elastic regime and one longitudinal hydro- caveat that U can include an additional temperature-dependent dynamic mode with frequency below ωF (shear mode is non- term due to cooperativity of molecular relaxation, in which propagating below frequency ωF as discussed below). case τ grows faster-than Arrhenius (‘super-Arrhenius’) as dis- Recall Landau’s assertion that a thermodynamic theory of cussed below. This cooperative process is of the same nature as liquids can not be developed because liquids have no small the one governing the non-exponentiality of relaxation in (16). parameter. How is this fundamental problem addressed here? We recall the result from the non-linear theory that a According to Frenkel reduction, liquids behave like solids bifurcation emerges when the energy of the particle becomes with small oscillating particle displacements serving as a small comparable to the coupling energy between two non-linear parameter. Large-amplitude diffusive particle jumps continue oscillators. Noting that this result is derived approximately, we to play an important role, but do not destroy the existence of can relate the coupling energy to the activation energy given in the small parameter. Instead, the jumps serve to modify the (10). Indeed, the coupling energy in the system of non-linear phonon spectrum: their frequency, ωF, sets the minimal fre- equations is the energy that a particle needs to escape a bound quency above which the small-parameter description applies state with another particle. This energy is of the same nature and solid-like modes propagate. and order of magnitude as that needed to break the atomic cage This approach is therefore a method of non-perturba- shown in figure 5. Therefore, the approximation in the Frenkel tive treatment of strong interactions, the central problem in theory is of the same nature as the one in the non-linear theory. field theories and other areas of physics [45]. It is markedly

9 Rep. Prog. Phys. 79 (2016) 016502 Review

different from any other method of treating strong interactions relaxation process below Tg, and remain to be of unknown nature contemplated in areas outside of liquids. [34, 35, 51–55]. An example of the super-Arrhenius behaviour is shown in figure 6 for a commonly measured glass-forming system, Continuity of solid and liquid states salol [56]. Here and in other cases, τ is described by the VFT and liquid-glass transition dependence fairly well, although a more careful experimental analysis revealed that on lowering the temperature, τ crosses The picture of liquid based on relaxation time τ has a notable over from the VFT to Arrhenius (or nearly Arrhenius) behav- consequence for liquid-solid transitions. In 1935, Frenkel ior [57–62]. This takes place at about midway of the glass published an article in Nature entitled, Continuity of the solid −6 ‘ transformation range where τ ≈ 10 s, i.e. above Tg and hence and the liquid states , [48] where he proposed and later devel- ’ well above T0. Known more widely in the experimental com- oped [1] an argument that liquids and solids are qualitatively munity as compared to theorists, the crossover removes the the same. This follows from the concept of τ: as τ increases basis for considering divergences and a possible thermody- on lowering the temperature beyond the experimental time namic phase transition at T0. frame, the liquid becomes frozen glass, and supports shear An interesting question is what causes the crossover from modes at all frequencies including at zero frequency. Hence, the VFT law at high temperature to nearly Arrhenius at low. liquids and solids are different in terms of τ only, i.e. quan- A useful insight comes from the observation that a sudden titatively, but not qualitatively. Frenkel subsequently stated local jump event such as the one shown in figure 5 induces that ‘classification of condensed bodied into solids and liquids an elastic wave with a wavelength comparable to interatomic (has) a relative meaning convenient for practical purposes but separation and cage size. This wave propagates in the system devoid of scientific value’ [1], an assertion that many would and affects relaxation of other events, setting the cooperativity find unusual today let alone then. of molecular relaxation. As discussed in the next section, being This idea was quickly criticized by Landau [49, 50] on a high-frequency wave, it propagates in the solid-like elastic the basis that the liquid-crystal transition involves symmetry regime with the propagation length given by equation (26): changes and therefore can not be continuous according to the phase transitions theory. This debate unfortunately reflected d =≈λωτ cτ (19) a misunderstanding because Frenkel was emphasizing super- where c is the speed of sound, ω is frequency and λ is wave- cooled liquids that becomes glasses on cooling, rather than length. As discussed in the next section, d increases with τ in crystals [1]. this regime, in contrast to the propagation length of the com- Remarkably, essentially the same debate is still continuing monly considered hydrodynamic waves [3]. in the area of liquid-glass transition where one of the main At high temperature when ττ≈ D, d =≈caτD , where discussed questions is whether a phase transition is involved a is interatomic separation. This means that the wave does not [34, 35, 51–55]? According to the large set of experimental propagate beyond the nearest neighbors and that the relaxa- data, liquids and glasses are structurally identical, and liquid- tion is non-cooperative (independent) and is Arrhenius and glass transition does not involve structural changes. Yet at exponential as a result. Importantly, d increases on lowering the glass transition temperature Tg the heat capacity changes the temperature because τ increases. This increases the coop- with a jump, seemingly providing support to the thermody- erativity of molecular relaxation [41] but only until d reaches namic signature of the glass transition. Here, Tg is defined as system size L. Therefore, the crossover from the VFT law to temperature at which τ exceeds the experimental time frame, Arrhenius takes place at τ = L, in quantitative agreement with τ = 102 103 s, corresponding to liquid becoming frozen in c – experiments [44]. terms of particle rearrangements during the observation period. We will return to the question of heat capacity jump at Tg when we discuss thermodynamic properties of viscous liq- Hydrodynamic and solid-like elastic regimes of uids. Here we note that although few consider the jump of wave propagation heat capacity at Tg as a phase transition, versatile proposals were related to a possible phase transition at lower tempera- As discussed above, liquids behave differently depending ture TT0g< [34, 35, 51–55]. The possibility of this was sug- on observation time or frequency. Frequencies ω > ωF and gested by the Vogel–Fulcher–Tammann (VFT) temperature ω < ωF correspond to solid-like elastic regime (ωτ > 1) and dependence of τ: hydrodynamic regime (ωτ < 1), respectively. The two regimes are described by different equations, those of elasticity [33] ⎛ A ⎞ and hydrodynamics [3]. The transition between the two τ ∝ exp⎜ ⎟ (18) ⎝ TT− 0 ⎠ regimes can be most easily seen by considering the response of the right-hand side of equation (12) to a periodic force where A and T0 are constants. PA= expi()ωt , giving τ in the VFT law diverges at T0, and the same applies to viscosity η according to equation (15). This led to propos- ⎛ 1 dPxy Pxy ⎞ 1 als that the ideal glass transition takes place at T to the ⎜ +=⎟ expi()ωtP()1i+ ωτ (20) ‘ ’ 0 ⎝ G dt η ⎠ η ideal glass state. The transition and the state are ostensibly not seen because its observation is suppressed by very slow where we used η = Gτ.

10 Rep. Prog. Phys. 79 (2016) 016502 Review

11P dP G For ωτ > 1, (20) gives ()iiωτ P ==ω , or purely M η GGdt = 1 (25) 1 + elastic response. For ωτ < 1, (20) returns purely viscous iωτ response, P. η If M = R expi()φ , the inverse complex velocity is To discuss liquid’s ability to operate in both regimes 1 ==ρρcosiφφ− sin , where ρ is density. P and s depending on ωF, we can either start with hydrodynamic equa- v MR()22 tions and introduce the solid-like elastic response or start with depend on time and position x as ft=−expi((ω xv/ )). Using elasticity equations and introduce the hydrodynamic response. the above expression for 1, ft=−expiωβexpikx exp − x , v () ()() The first method has received most attention in the history of ρ φ where k = ω ρ cos φ and absorbtion coefficientβω = sin . liquid research, and generally forms the basis for a variety of R 2 R 2 Combining the last two expressions for k and , we write approaches collectively known as ‘generalized hydrodynam- β 2tπ an φ ics’ discussed in the later section. The second method is not β = 2 , where λ = 2π is the wavelength. commonly discussed and its implications are less understood. λ k From equation (25), tan φ = 1 . For high-frequency waves Below we consider important examples of the difference ωτ in which collective modes operate in the hydrodynamic and ωτ 1, tan φφ≈=1 , giving β = π . Lets introduce the solid-like elastic regimes, and start with the second method. ωτ λωτ propagation length d = 1/β so that fx∝−exp/()d . Then, d = λωτ . Therefore, this theory gives propagating shear waves π Modifying elasticity: including hydrodynamics in elasticity in the solid-like elastic regime ωτ 1, with the propagation equations length Condition (11), ωτ > 1 (sometimes written as ωτ 1) corre- d ≈⋅λωτ sponds to wave propagation in the liquid with frozen structure (26) (as in a solid), where the microscopic equations are Newton ()ωτ 1 equations for all particles (3) and (4). This is the solid-like We note that this result is derived for plane waves, and it elastic regime of wave propagation. Modifying elasticity approximately holds in disordered systems for wavelengths equations by including hydrodynamics enables us to address that are large enough. At smaller wavelengths comparable to our first case study, the difference of wave propagation in structural inhomogeneities, d is reduced due to the dissipation regimes ωτ > 1 and ωτ < 1. We will see that dissipation of plane waves in the disordered medium. The dissipation is length, the length over which an induced wave is dissipated related to how well the eigenstates of the disordered system due to viscous effects, behaves qualitatively differently in the can be approximated by plane waves (for more detailed dis- two regimes. cussion, see ([46]). We consider both elastic and viscous response in the form In the hydrodynamic regime ωτ 1, we findφ = π and equivalent to equation (12) 2 d = λ . Different from the high-frequency case, this means ds P 1 dP 2π =+ (21) that low-frequency shear waves are not propagating (because d2t η 2G dt they are dissipated over the distance comparable to the wave- where s is shear strain and introduce the operator length), a result that is also known from hydrodynamics [3]. The consideration of the propagation velocity of longitudi- d A =+1 τ (22) nal waves involves the bulk modulus which can be written in dt K2 the form LK=+1 containing the non-zero static part 1 + 1 where τ = η from (15). Then, equation (21) can be written as iωτ G as well as the frequency-dependent part as in (25). Repeating ds 1 the same steps as above, the propagation length in the solid- = AP (23) dt 2η like elastic regime ωτ 1 is the same as in equation (26). In the hydrodynamic regime ωτ 1, the propagation length If A−1 is the reciprocal operator to A, PA= 2η −1 ds. Because becomes dt d = A − 1 from equation (22), P = 2G(1 − A−1)s. Comparing dt τ λ this with P = 2Gs, we find that the presence of relaxation d ≈ ωτ process is equivalent to the substitution of G by the operator ()ωτ 1 (27) M = G(1 − A−1). The above constitutes the modification of the constituent Comparing equations (26) and (27), we see that the two elasticity equations by introducing the relaxation process in the different regimes give qualitatively different character of liquid and τ, i.e. approach to liquids from the solid elastic state: waves dissipation: the propagation length increases with τ

−1 and viscosity in the former, but decreases with τ and viscosity PG==22sP→(GA1 − )s (24) in the latter. Let us now consider the propagation of the wave of P and s The decrease of the propagation length with liquid vis- with time dependence expi()ωt . Differentiation gives multipli- cosity in the commonly discussed hydrodynamic regime is a cation by iω. Then, A =+1iωτ, and M is: familiar result from fluid mechanics [3]. On the other hand,

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the increase of propagation length in the solid-like elastic Equations (22)–(24) account for both long-time viscosity regime is less known. and short-time elasticity. From (21)–(23), we see that account- An important insight from this discussion is that the two ing for both effects is equivalent to making the substitution regimes of waves propagation are different from the physical 111d → + . Using η = Gτ from equation (15), the substitu- point of view and yield qualitatively different results, includ- ηηGtd ing directly opposite results for the propagation length. This tion becomes: implies that essential physics in the hydrodynamic regime and 11⎛ d ⎞ its underlying equations can not be extrapolated to the solid- → ⎜⎟1 + τ (29) ηη⎝ dt ⎠ like elastic regime (and vice versa). By extrapolating here we mean extending the hydrodynamic regime to large k and ω Using (29) in equation (28) gives while keeping the underlying physics and associated equa- 2 ⎛ d ⎞⎛ dv ⎞ tions qualitatively the same. We will return to this point below η∇=v ⎜⎟1 ++τρ⎜⎟∇p (30) when we discuss the approach to liquids based on generalized ⎝ dtt⎠⎝ d ⎠ hydrodynamics. Having proposed equation (30), Frenkel did not analyze it Our second case study is related to the crossover between or its solutions. We do it below. two regimes of propagation. In the solid-like elastic regime, the We consider the absence of external forces, p = 0 and the BG+ 4 slowly-flowing fluid so that d = ∂ . Then, equation (30) reads propagation velocity in the isotropic medium is v = 3 dtt∂ ρ [33], where B and G are bulk and shear moduli, respectively. ∂2v ∂2v ∂v η = ρτ + ρ (31) This is the case for solids as well as liquids in the solid-like ∂x2 ∂t2 ∂t elastic regime where shear waves above ωF are propagating. In the hydrodynamic regime where no shear waves propagate where v can be y or z velocity components perpendicular to x. In contrast to the Navier Stokes equation, the general- B – as discussed above, the propagation speed is v = , corre- ized hydrodynamic equation equation (31) contains the sec- ρ sponding to G = 0. Therefore, Frenkel argued, the transition ond time derivative of v and hence allows for propagating between the two regimes results in the noticeable increase of waves. Indeed, equation (31) without the last term reduces to the wave equation for propagating shear waves with velocity the propagation speed by a factor 1 + 4 G . The transition can 3 B η G cs ==. The last term represents dissipation. Using be achieved by either changing τ at a given frequency by alter- τρ ρ 2 ing temperature or pressure, or by changing frequency at fixed η ==Gcτρsτ, we re-write equation (31) as temperature and pressure. ∂2v ∂2v 1 ∂v In the later section, fast sound , we will revisit this effect c2 ‘ ’ s 2 = 2 + (32) on the basis of recent experimental results. ∂x ∂t τ ∂t Seeking the solution of (32) as vv=−0 expi((kx Ωt)) gives the quadratic equation for Ω: Modifying hydrodynamics: including elasticity 2 i 2 2 in hydrodynamic equations Ω+Ω−cks = 0 (33) τ Equations (22)–(24) modify (generalize) elasticity equa- 1 Equation (33) has purely imaginary roots if csk < , approx- tions by including relaxation and viscous effects in the 2τ liquid in the form of viscous flow at times longer thanτ . imately corresponding to the hydrodynamic regime ωτ < 1. Equally, Frenkel argued [1], one can generalize hydrody- Therefore, we find that shear waves are not propagating in the namic equations by endowing the system with the solid- hydrodynamic regime ωτ < 1, which is the same result as the like property to sustain shear stress at times shorter than τ. one derived in the previous section where elasticity equa- This idea is generally similar in its spirit to the approach of tions were modified to include viscous effects. 1 generalized hydrodynamics that appeared later (see gen- If csk > (corresponding to the solid-like elastic regime ‘ 2τ eralized hydrodynamics’ section below), although Frenkel i 2 2 1 ωτ > 1), equation (33) gives Ω=−±cks − 2 , and we implemented the idea differently. Apart from the general 2τ 4τ find interest, this implementation deserves attention because it is not discussed in traditional generalized hydrodynamics ⎛ t ⎞ v ∝−exp⎜⎟expi()ωt approaches [5, 7, 8]. ⎝ 2τ ⎠ Lets write the Navier–Stokes equation as 2 2 1 ω =−cks (34) 4τ2 2 1d⎛ v ⎞ ∇ v =+⎜⎟ρ ∇p (28) η ⎝ dt ⎠ Equation (34) describes propagating shear waves, contrary to the original Navier–Stokes equation. We therefore where v is velocity, p is pressure, η is shear viscosity, ρ is den- find that shear waves are propagating in the solid-like elas- sity and the full derivative is d =+∂ v∇. dtt∂ tic regime ωτ > 1, the same result we derived in the previous

12 Rep. Prog. Phys. 79 (2016) 016502 Review section where elasticity equations were modified to incorpo- Experimental evidence for high-frequency rate fluidity. collective modes in liquids According to equation (34), the increase of τ or viscosity gives smaller wave dissipation (larger lifetime) in the solid- Low-frequency collective modes, including familiar sound like elastic regime ωτ > 1, contrary to the hydrodynamic waves, are well understood in liquids. Yet these modes make regime [3]. This is the same effect that we have discussed a negligible contribution to liquid energy and heat capacity. in the previous section where we found the increase of the Indeed, the liquid energy is almost entirely governed by high- propagation length of shear waves with τ and viscosity (see frequency modes due to the approximately quadratic density equation (26)). of phonon states. However, the prediction of high-frequency We note that for large τ or viscosity, ω in equation (34) solid-like modes in liquids in the regime ωτ > 1 was non-triv- becomes ω = cks as in the case of shear waves with no dis- ial, and was outside the commonly discussed hydrodynamic sipation at all. These are solid-like elastic waves with wave- approach where ωτ < 1 [2–12]. lengths extending to the shortest interatomic separations and The experimental evidence supporting the propagation frequencies up to the highest Debye frequency as predicted in of high-frequency modes in liquids includes inelastic x-ray, the solid-like elastic approach by equation (11). neutron and Brillouin scattering experiments. Most of the evi- We also note that ω of shear waves in equation (34) does dence is recent and follows the deployment of powerful syn- not increase from 0 to its linear branch ω = cks in a jump- chrotron sources of x-rays. like manner as follows from (11). Instead, starting from about Early experiments detected the presence of high-frequency 1 propagating modes and mapped dispersion curves which were ω ==ωF , ω gradually increases from the square-root τ in striking resemblance to those in solids [63]. This and simi- dependence to the linear dependence ω = cks at large τ. This lar results were generated at temperature around melting. The is consistent with the experimental result showing a gradual measurements were later extended to high temperatures con- increase of the speed of sound and shear rigidity with the wave siderably above the melting point, confirming the same result. frequency [85]. We will revisit this point when we discuss the It is now well established that liquids sustain propagating phonon approach to liquid thermodynamics. modes extending to wavelengths comparable to interatomic To derive the propagation of longitudinal waves, we need separations [64–76]. More recently, the same result has been to include the longitudinal viscosity in the Navier–Stokes asserted for supercritical fluids [67, 74, 75]. equations and modify it similarly to (29), remembering that Importantly, the propagating modes in liquids include bulk viscosity is related to the bulk modulus which, in addi- transverse modes. Initially detected in highly viscous liquids tion to frequency-depending term, always has non-zero static (see, e.g. [77, 78]), transverse modes have been later studied in term [1]. This will give propagating longitudinal waves in both low-viscous liquids on the basis of positive dispersion [64–66, solid-like elastic regime and in the hydrodynamic regime, in 68] (recall our previous discussion that the presence of high- agreement with the results in the previous section. We will not frequency transverse modes increases sound velocity from the pursue this derivation here. hydrodynamic to the solid-like value). These studies included Therefore, we find that as far as wave propagation is con- water [79], where it was found that the onset of transverse cerned, equations of hydrodynamics modified (generalized) to excitations coincides with the inverse of liquid relaxation time include solid-like elastic effects give the same results as equa- [80], as predicted by (11). More recently, transverse modes in tions of elasticity modified to include viscous effects. liquids were directly measured in the form of distinct disper- Interestingly, it is the approach of ‘generalized hydrody- sion branches and verified on the basis of computer modeling namics’ which historically received wide attention and devel- [69–73]. opment and has become a distinct area of research [5, 7, 8]. In figure 7, we show measured dispersion curves measured We will discuss this approach in the later section. This reflects in liquid Na [70] and liquid Ga [71], together with SiO2 glass the historical trend we alluded to in the introduction: the com- [81, 82] for comparison. In figure 8, we show the dispersion munity largely viewed liquids as systems conforming to the curves recently measured in liquid Sn [72], Fe, Cu and Zn hydrodynamic equation at the fundamental level, with pos- using the experimental setup to study liquids with high melt- sible solid-like elastic effects to be introduced, if needed, on ing points [73]. top. To some extent, this view was consistent with existing In figure 7, we observe a striking similarity between experiments at the time that mostly probed low-energy prop- liquids and their polycrystalline and crystalline counterparts erties of liquids. As discussed in the next section, high-energy in terms of longitudinal and transverse dispersion curves. We experiments uncovering solid-like properties of liquids have further note the similarity of dispersion curves in liquids and emerged relatively recently. solid glasses. Overall, figures 7 and 8 present an important It can be argued that the approach to liquids starting with the experimental evidence regarding collective excitations in solid-like elastic description contains more information about liquids. We observe that despite topological and dynamical structure and dynamics and, therefore, is more suited to discuss disorder, solid-like quasi-linear dispersion curves exist in high-frequency dynamics of liquids. This becomes particularly liquids in a wide range of k and up to the largest k correspond- important for constructing the phonon theory of liquid thermo- ing to interatomic separations, as is the case in solids. Notably, dynamics where high-frequency modes govern system’s energy this includes both high-frequency longitudinal and transverse and heat capacity as discussed in the later section. modes.

13 Rep. Prog. Phys. 79 (2016) 016502 Review experimental dispersion curves obtained by harmonic probes such as x-rays or neutrons show that high-frequency plane waves are propagating in liquids, as witnessed by the data in figures 7 and 8. From the physical point of view, this follows from the fact that despite long-range disorder, a well-defined short-range order exists in liquids, glasses and other disordered systems, as is seen from the peaks of pair distribution functions in the short as well as medium range. Therefore, high-frequency harmonic plane waves, even though damped, are able to propagate at least the distance comparable to the typical length of the short-range order. We will find below that this length, the interatomic separation, which is also the fundamental length of the system, plays a profound role in governing the thermodynamic properties of liquids. We have noted the similarity of vibrational properties between disordered liquids and their crystalline counterparts. Interestingly, similarity (and the lack thereof) between disordered glasses and their parent crystals have also been widely discussed. The widely discussed ‘Boson’ peak in the low-frequency range has been long thought to be present in glasses only but not in crystals and to originate from disorder. However, later work [83, 84] has demonstrated that similar vibrational features are present in crystals as well, provided glasses and crystals have similar density.

Fast sound

It is now good time to revisit the origin of fast sound mentioned earlier using detailed experimental data discussed in the previous section. Starting from larger k-values, the measured speed of sound often exceeds the hydrodynamic value. This is seen in figure 8 where the hydrodynamic speed of sound is shown as a dashed line. The increase of the measured speed of sound over its Figure 7. Experimental dispersion curves. (a): longitudinal hydrodynamic value is often called as ‘fast sound’ or ‘positive (filled black bullets) and transverse (filled red bullets) dispersion sound dispersion’ (PSD). curves in SiO2 glass [81]. (b): longitudinal (filled black bullets) and transverse (filled red bullets) excitations in liquid Na. Open We recall Frenkel prediction discussed earlier: at high diamonds correspond to longitudinal (black) and transverse (red) frequency where liquid’s shear modulus becomes non-zero, excitations in polycrystalline Na, and dashed–dotted lines to the propagation velocity crosses over from its hydrodynamic longitudinal (black) and transverse (red) branches along [1 1 1] 4 B BG+ direction in Na single crystal [70]. (c): longitudinal (black bullets) value v = to the solid-like elastic value v = 3 (see, ρ ρ and transverse (red bullets) excitation in liquid Ga. The bullets are e.g., [33, 34]) where B and G are bulk and shear moduli, bracketed by the highest and lowest frequency branches measured in bulk crystalline β-Ga along high symmetry directions, with black respectively. and red dashed-dotted lines corresponding to longitudinal and The physical origin of the fast sound has remained con- transverse excitations, respectively [71]. Dispersion curves in Na troversial, including understanding relative contributions and Ga are reported in reduced zone units. of the above mechanism and other effects such as disorder. Experimentally, the crossover of the longitudinal sound veloc- We comment on damping of collective modes in liquids. ity from its hydrodynamic to solid-like elastic value has been A conservative system, crystalline or amorphous, has its been well-studied in viscous liquids where the system starts eigenmodes which are non-decaying. Indeed, equation (4) sustaining rigidity at MHz frequencies (see, e.g. [85], where does not require system’s crystallinity. For a disordered struc- fast sound is seen at fairly large wavelengths at which the ture, equation (4) gives eigenstates and eigenfrequencies cor- liquid can be considered as a homogeneous medium). It is responding to collective non-decaying excitations. For long generally agreed that in this range of frequencies, fast sound wavelengths and small energies, these states are similar to har- originates from this mechanism [85]. monic plane waves and their damping in disordered systems is At smaller wavelengths approaching the length of medium small. For short wavelengths, the eigenstates of the disordered and short-range order, the wave feels structural inhomoge- system are different from the plane waves, and so damping of neities, and disorder of liquids and glasses starts to affect the short-wavelength plane waves becomes appreciable. Yet the dispersion relationship. PSD, with the relative magnitude of

14 Rep. Prog. Phys. 79 (2016) 016502 Review

Table 1. Comparison of experimental vl and vl calculated on the basis of vh and vt using equation (35) as discussed in the text.

vl (experimental) vl (calculated) −1 −1 −1 −1 vh (m s ) vt (m s ) (m s ) (m s )

Fe 3800 1870 ± 50 4370 ± 30 4370 ± 50 Cu 3460 1510 ± 50 3890 ± 30 3875 ± 50 Zn 2780 1620 ± 50 3330 ± 30 3350 ± 50 Sn 2440 1220 ± 150 2890 ± 30 2820 ± 150

Note: The data for vl, vt and vh is from [72] and [73].

related to the short-range order in disordered systems [76]), although large PSD in silica glass may be related to the effect of mixing with the low-lying optic modes. In water, fast sound was discussed on the basis of coupling between the longitudinal and transverse excitations, and it was found that the onset of transverse excitations coincides with the inverse of liquid relaxation time [79, 80], as predicted by (11). Recent detailed experimental data discussed in the previous section enable us to directly address the origin of the fast B G sound and its magnitude. Combining vh = , vt = and ρ ρ BG+ 4 3 vl = (see, e.g., [34]), where vh is the velocity of the ρ low-frequency hydrodynamic sound, vt is the transverse sound

velocity and vl is the longitudinal velocity from the measured dispersion curves, we write 4 vv2 =+2 v2 (35) l h 3 t We note that the expression vv2 =+B 4 2 is the identity l ρ 3 t for isotropic solids, and also applies to liquids in which the longitudinal speed of sound changes from the hydrodynamic to solid-like elastic value due to the onset of shear rigidity. Using the data from [72] and [73], we have taken vl and vt from the dispersion curves for Fe, Cu, Zn and Sn shown in figure 8 at k points where the observed PSD is maximal and where ω()k (E(k)) is in the quasi-linear regime before starting to curve at large k. For Fe, Cu, Zn, we use the new data shown in blue in figure 8 and consider the following k points: −1 Figure 8. Longitudinal (black and blue crosses) and transverse k = 7.7 nm (first point on the transverse branch in figure 8), (filled red bullets) dispersion curves in (a) liquid Sn [72], (b) liquid k = 7.8 nm−1 (second point on the transverse branch) and Fe, (c) liquid Cu and (d) liquid Zn [73]. Red filled triangles in k = 8 nm−1 (second point on the transverse branch), respec- (a) are the results from ab initio simulations [72]. Blue and black tively. For Sn, large PSD is seen at about k = 3.3 nm−1 cor- crosses correspond to recent and earlier experiments, respectively. Dashed lines indicate the slope corresponding to the hydrodynamic responding to the second point on the longitudinal branch in sound in the limit of low k. figure 8(a). To find vt at this k, we extrapolated the higher- lying transverse points to lower k while keeping them parallel −1 to the simulation points, yielding vt =±1220 150 m s . few per cent, was observed in a model harmonic glass and Using experimental vh and vt, we have calculated vl using attributed to the ‘instantaneous relaxation’ due to fast decay equation (35). We show calculated and experimental vl in and dissipation of short-wavelength phonons in a disordered table 1 below. system [86]. Later work demonstrated that starting from mes- We observe in table 1 that the calculated and experimental oscopic wavelengths, the effective speed of the longitudinal vl agree with each other very well. We therefore find that the sound can also decrease [87–89]. Different mechanisms and mechanism of fast sound based on the onset of shear rigidity contributions to PSD were subsequently discussed [76, 90]. quantitatively accounts for the experimental data of real liq- The instantaneous relaxation is likely to be significant close uids in the wide range of k spanning more than half of the first to the zone boundary [81] (or the first Brillouin pseudo-zone, Brillouin pseudo-zone.

15 Rep. Prog. Phys. 79 (2016) 016502 Review

It is interesting to discuss pressure and temperature con- IR peak, IBM, is the Landau–Placzek ratio: =−γ 1. Applied ditions at which the fast sound operates in this picture. The IBM originally to light scattering experiments, equation (36) is also above mechanism implies that the fast sound disappears when viewed as a convenient fit to high-energy experiments probing the system loses shear resistance and transverse modes at non-hydrodynamic processes where the fit that may include all available frequencies. As discussed later, this takes place several Lorentzians or their modifications. above the Frenkel line which demarcates liquid-like and gas- Generalizing hydrodynamic equations and extending like properties at high temperature including in the supercriti- them to large k and ω is often done in terms of correlation cal region. functions. Solving the hydrodynamic Navier Stokes equa- As already mentioned, other effects contributing to PSD – tion for the transverse current correlation function Jkt, , can be operative, including the effects due to disorder at t() ∂ Jkt,,=−νk2Jkt, where ν is kinematic viscosity, gives for large k. ∂t t() t() the Fourier transform Jkt(), ω a Lorentzian form similar to (36): 2 Generalized hydrodynamics 2 νk Jkt ,2ω = v () 0 222 (37) ων+ ()k In the earlier section, we have discussed modifying (general- 2 izing) hydrodynamic equations by including solid-like elastic where ν is kinematic viscosity and v0 ==Jktt(),0. effects as one way to describe both elastic and hydrodynamic The generalization is done in terms of the memory function K (k, t) defined in the equation for Jk, ω as response of the liquid. ‘Generalized hydrodynamics’ as a dis- t t() tinct term refers to a number of proposals seeking to achieve t ∂ 2 essentially the same result by using a number of different phe- Jkt(),,ω =−kKt()kt− t′′Jktt(),dt′ (38) ∂t ∫0 nomenological approaches [5, 7, 8]. One starts with hydrodynamic equations initially applicable to low ω and k, and where Kt()kt, − t′ is the shear viscosity function or the mem- introduces a way to extend them to include the range of large ory function for Jkt(), ω which describes its time dependence ω and k. (‘memory’). From the point of view of thermodynamics, accounting for Introducing Jks˜t(), as the Laplace transform modes with high ω is important because these modes make the Jktt(),2ω = Re[Jks˜(), ]s=iω and taking the Laplace transform largest contribution to the system energy. The contribution of of (38) gives hydrodynamic modes is negligible by comparison. 2 1 Generalized hydrodynamics is a large field (see, e.g. Jks˜t(), = v0 (39) sk+ 2Kk˜ , s [5, 7, 8, 12]) which we can only discuss briefly emphasizing t() key starting equations and schemes of their modification to The generalization introduces the dependence k and ω include higher-energy effects, with the aim to offer readers a by writing K˜t()ks, as the sum of real and imaginary parts feel for methods used and physics discussed. [Kk˜ts(),,sK](=iω =+′t kKωω)(i,t″ k ). Then, The hydrodynamic description starts with viewing the 2 2 kK′t()k, ω liquid as a continuous homogeneous medium and constrain- Jkt ,2ω = v () 0 2 2 2 2 (40) ing it with continuity equation and conservation laws such ((ωω++kKt″ kk,,)) ((Kk′t ω)) as energy and momentum conservation. Accounting for ther- giving the generalized hydrodynamic description of the trans- mal conductivity and viscous dissipation using the Navier– verse current correlation function with a resonance spectrum. Stokes equation, the system of equations can be linearized Further analysis depends on the form of Kt(k, t), which is and solved. This gives several dissipative modes, from which often postulated as the evaluation of the density-density correlation function gives the structure factor S()k, ω in the Landau–Placzek form ⎛ t ⎞ Ktt()kt,,=−Kk()0exp⎜ ⎟ (41) which includes several Lorentzians: ⎝ τ()k ⎠ γ − 1 2χk2 1 Equation (41) decays with time relaxation time τ, and we Sk, ω ∝ + () 222 recognize that this is essentially the same behavior described γ ωχ+ ()k γ by earlier equations (14) or (16), except the postulated form 2 2 ⎛ Γk Γk ⎞ also assumes k-dependence of τ. In generalized hydrody- × ⎜ + ⎟ 222 222 ⎝ ()ωω++ck ()Γk ()−+ck ()Γk ⎠ namics, equation (41) is used not only for K but also for several types of correlation and memory functions. These (36) C often include modifications such as including more exponen- where χ is thermal diffusivity, γ = p and dissipation Γ Cv tials with different decay times in order to improve the fit to depends on χ, γ, viscosity and density. experimental or simulation data. The first term corresponds to the central Rayleigh peak and Mode-coupling schemes consider correlation functions for thermal diffusivity mode. The second two terms correspond to density and current density, factorise higher-order correlation the Brillouin–Mandelstam peaks, and describe acoustic modes functions by expressing them as the product of two time cor- with the adiabatic speed of sound c. The ratio between the inten- relation functions with coupling coefficients in the form of sity of the Rayleigh peak, IR, and the Brillouin–Mandelstam static correlation functions, and give a better agreement for

16 Rep. Prog. Phys. 79 (2016) 016502 Review the relaxation function as compared to the single exponential giving a finite static restoring force for the longitudinal mode. decay model. As in the previous considerations, this gives propagating lon- Neglecting k-dependence of τ for the moment, taking the gitudinal modes in the hydrodynamic regime. Rather than Laplace transforms of (41) to findK ′t()k, ω and Kt″()k, ω and postulating the relaxation functions Mt()kt, and Ml()kt, as in using them in (40) gives Jkt(), ω as [5] (41), the mode-coupling theory considers higher-order correlation functions and approximates them by the products of 1 Jk, two-time correlation functions. Memory functions can then be t()ω ∝ 2 22 1 (42) calculated using the results from molecular dynamics simula- ωτ−−kKtt()kf,0 2 + ((,,Kk0)) ()()2τ tions such as static correlation functions and other parameters required as the input. For simple systems, the onset of shear where f is the non-essential function of τ and K (k, 0). t wave propagation can be related to certain shoulder-like fea- The resonance frequency in (42) corresponds to the propa- tures in the calculated memory function. 2 1 gation of shear modes provided k Kkt ,0 > . This condi- ()2τ2 The amount of current research in generalized hydrody- tion defines the high-frequency regime of wave propagation namics has markedly decreased as compared to several dec- in the solid-like elastic medium. Importantly, this condition is ades ago [5]. Interestingly, the steer towards going beyond the essentially the same as the one we derived from the general- hydrodynamic description and generalized hydrodynamics ized hydrodynamic equation (32), as follows from the discus- came from the experimental, and not theoretical, community sion between equations (32) and (34). after the solid-like properties of liquids were discovered and Similar expressions can be derived for the longitudi- problems related to the hydrodynamic description of those nal current correlation function which also includes a static properties became apparent [66, 70]. Some of the more recent time-independent term which does not decay. This term cor- examples include exploring how hydrodynamic descrip- responds to non-zero bulk modulus which gives propagating tion gives rise to a single underlying relaxation process and longitudinal waves in the hydrodynamic regime, as discussed accounting for the viscoelastic effects using several first fre- in the previous section. quency moments (see [91–93] and references therein). Other An alternative approach to generalize hydrodynamics is to approaches assume ad hoc that more dynamical variables and make a phenomenological assumption that a dynamical vari- their second and third derivatives are involved in extrapolat- able in the liquid is described by the generalized Langevin ing the hydrodynamic regime to high k and ω [94] and, fol- equation: lowing earlier proposals [95], use the generalized collective modes schemes where the sum of exponentials such as (41) ∂at t ()+Ωidat +−at′′Kt tt′ = ft is assumed to describe the decay of correlations. General dis- () ∫0 () () () (43) ∂t advantages of this and similar schemes are related to the phe- where the first two terms reflect the possibility of propagat- nomenological and empirical nature of the method [66, 70]. ing modes, the third term plays the role of friction with the Continuing interest in generalized hydrodynamics is stimu- memory function K and f is the random force. lated by fitting the experimental spectra where, for example, the This approach proceeds by treating a(t) not as a single second-order memory function is assumed to take the exponen- variable but as a collection of variables of choice so that a(t) tial form (41) [76] or as a sum of two or more exponentials [96]. becomes a vector including, in its simplest forms, conserved density, current density and energy variables. These variables are further generalized to include their dependence on wave- Comment on the hydrodynamic approach to liquids number k. This gives a set of coupled equations solved in the matrix form. The set of dynamical variables can be extended Challenges involved in generalized hydrodynamics were to include the stress tensor and heat currents. In this case, the appreciated by practitioners at the early stages of develop- generalized viscosity is found to have the same exponential ment [5], including often phenomenological and empirical decay as in (41) once the stress tensor is explicitly introduced ways involved in extrapolating hydrodynamic description into as a dynamical variable, the assumption is made regarding the solid-like elastic regime. We do not review these here, stress correlation function and a number of approximations although we note the following. Generalized hydrodynamics are made. Then, similar viscoelastic effects are found as in the introduces k and ω-dependencies in the liquid properties such previous approach [5]. as diffusion, viscosity, thermal conductivity, heat capacity and Propagation of shear and longitudinal modes is also dis- so on, with the aim to calculate and discuss these functions in cussed in the mode-coupling theories mentioned above. The the non-hydrodynamic regime. It is not entirely clear what is theory seeks to take a more general approach in the follow- the physical meaning of concepts such as diffusion or viscos- ing sense. Considering that correlation functions are due ity at large ω where the system’s response is elastic rather than to density and current density correlators, the theory repre- viscous. Understanding physical effects at these frequencies is important because short-wavelength modes govern most sents K˜t()ks, in (39) by the second-order memory functions important system properties such as energy. Mt()kt, and Ml()kt, for transverse and longitudinal currents, We question a more fundamental premise of the hydro- so that the transverse function Jks˜t(), and longitudinal func- dynamic description of liquids: the advantage of approach- tion Jk˜l(), s acquire the forms of damped oscillators. Jk˜l(), s ‘ ing the large (k, ) region by generalizing the hydrodynamic differs from Jks˜t(), by the presence of non-zero static term, ω

17 Rep. Prog. Phys. 79 (2016) 016502 Review description is that one maintains contact with the long-wave- In equation (44), K is the sum of all kinetic terms includ- length, low-frequency region at all stages of the development. ing vibrational and diffusional components. In the classical This gives insight to the structure of the resulting equation [5]. 3 ’ case, K = NkBT, and does not depend on how the kinetic Although being able to track the evolution of equations may 2 energy partitions into oscillating and diffusive components. be insightful in some cases, it may not be advantageous in gen- P and P ωω> are potential energies of the longitudinal eral. There is no fundamental reason to designate the hydrody- l t()F mode and transverse phonons with frequency ω > ω , respec- namic approach as the universally correct starting point. The F tively. For now, we tentatively include in equation (44) the traditional reason for the hydrodynamic approach to liquids term P , related to the energy of interaction of diffusing parti- is that they are flowing systems and therefore obey hydrody- d cles with other parts of the system. P is understood to be part namic equations. As we have discussed above, this applies d of system’s potential energy which is not already contained in to times t > τ (ω < ω ) only whereas for t < τ (ω > ω ) the F F the potential energy of the phonon terms, P and P ωω> . system is solid-like and can be described by solid-like equa- l t()F P is small compared to other terms in (44) as discussed below. tions. Furthermore, we have seen that the same properties of d The smallness of P can be discussed by approaching the collective modes are obtained by either starting with the d liquid from either gas or solid state. Lets consider a dilute inter- hydrodynamic equations and incorporating solid-like elastic acting gas where system s potential energy is entirely given effects or starting with the elasticity equations and incorporat- ’ by the potential energy of the longitudinal mode, P , with the ing the hydrodynamic fluidity. l available wavelengths that depend on pressure and tempera- Instead, we propose that for the purposes of fundamen- ture. The remaining energy in the system is the kinetic energy tal microscopic description, liquids should be considered corresponding to the free particle motion, giving P = 0. for what they are: systems with molecular dynamics of both d Density increase (and temperature decrease) result in decreas- types, solid-like oscillatory motion and diffusive jumps, with ing wavelength of the longitudinal mode until it reaches val- relative weights of these motions changing with tempera- ues comparable to solid-like interatomic separation a (see the ture. As discussed below, these relative weights govern most earlier section experimental evidence for high-frequency col- important system properties. In this approach, the hydrody- ‘ lective modes in liquids ). In this dense gas regime, the sys- namic regime ( 1) and solid-like elastic regime ( 1) ’ ωτ < ωτ > tem s potential energy is still given by P , which is the energy can, and in many cases should, be considered separately and ’ l of longitudinal mode but now with the full solid-like spectrum without necessarily seeking to extrapolate one regime onto the of wavelengths ranging from the system size to a. Further den- other. In addition to avoiding problems of ad-hoc extrapola- sity increase or temperature decrease result in the appearance tion assumptions often present in generalized hydrodynamics, of the solid-like oscillatory component of motion. This pro- this approach has the added benefit of rigorously delineat- cess is most conveniently discussed above the critical point ing different regimes of liquid dynamics where important where no liquid gas phase transition intervenes and where the properties are qualitatively different. This will become par- – crossover from purely diffusive motion to combined diffusive ticularly apparent when we discuss the change of dynamics in and solid-like oscillatory motion takes place at the Frenkel line the supercritical region at the Frenkel line, the effect that the discussed in later sections. The emergence of solid-like oscilla- hydrodynamic description misses. tory component of particle motion is related to the emergence The hydrodynamic and solid-like elastic descriptions of of transverse modes with frequency ω > ω in equation (44). liquids apply in their respective domains. It turns out that it is F The potential energy of transverse modes now contributes to the solid-like description that is relevant for constructing the the system s potential energy, and the remaining energy cor- thermodynamic theory of liquids discussed in the next section. ’ responds to the free particle motion (P = 0 in equation (44)) This is because high-frequency modes make the largest contri- d as in the dense gas. bution to the system energy due to quadratic density of states We can also approach the liquid from the solid state. In the 2, and propagate in the solid-like elastic regime 1. ∝ ω ωτ > solid, the potential energy is the sum of potential components of Importantly, this does not require extrapolations involved in longitudinal and transverse modes. The emergence of diffusive the generalized hydrodynamics approach. motion in the liquid results in the disappearance of transverse modes with frequency ω < ωF according to (11) and modifies Phonon theory of liquid thermodynamics the potential energy of transverse modes to Pt()ωω> F in equation (44). This implies smallness of low-frequency potential Harmonic theory energy of transverse modes: PPtt()ωω<>FF ()ωω, where We have seen above that collective modes in liquids include Pt()ωω< F is the potential energy of low-frequency trans- one longitudinal mode and two transverse modes propagat- verse modes. Instead of low-frequency transverse vibrations 1 with potential energy P in a solid, atoms in a liq- ing at frequency ω >=ωF in the solid-like elastic regime. t()ωω< F τ The energy of these modes is the liquid vibrational energy. In uid ‘slip’ and undergo diffusive motions with frequency ωF addition to oscillating, particles in the liquids undergo diffu- and associated potential energy Pd, hence PPdt≈<()ωωF . sive jumps between quasi-equilibrium positions as discussed Combining this with PPtt()ωω<>FF ()ωω, PPdt ()ωω> F above. We write the total liquid energy as follows. Re-phrasing this, were Pd large and comparable to Pt()ωω> F , strong restoring forces at low frequency E =+KPlt+>PP()ωωF + d (44) would result, and lead to the existence of low-frequency

18 Rep. Prog. Phys. 79 (2016) 016502 Review

vibrations instead of diffusion. We also note that because 1 where ωF = , ωli and ωti are frequencies of longitudinal and PP≈ , PP ωω> gives PP , further implying that P τ ltdt()F dl d transverse waves, N is the number of atoms and N′ is the num- can be omitted in equation (44). ber of phonon states that include longitudinal waves and trans- We note that in the regime ττ , the justification for D verse waves with frequency ω > ω . Here and below, k = 1. the smallness of P in the two previous paragraphs becomes F B d We recall our earlier discussion that the longitudinal mode unnecessary. Indeed, using a rigorous statistical-mechanical propagates in two different regimes: hydrodynamic regime argument it is easy to show that the total energy of diffus- ωτ < 1 or solid-like elastic regime ωτ > 1. This gives different ing atoms (the sum of their kinetic and potential energy) can dissipation laws in the two regimes, but this circumstance is be ignored to a very good approximation if ττ . This is D unimportant for calculating the energy. Indeed, (48) makes no explained in the viscous liquids section below in detail (see ‘ ’ reference to dissipation, and includes the mode energy and the equations (69), (70) and (73) and discussion around them), density of states only. These are the same in the two regimes, where we also remark that ττ corresponds to almost entire D and hence for the purposes of calculating the energy, the lon- range of τ in which liquids exist as such. gitudinal mode can be considered as one single mode with Neglecting small P in equation (44) is the only approxima- d Debye density states. This statement is not entirely correct tion in the theory; subsequent transformations serve to make because the mode is not well described in the regime ωτ ≈ 1, the calculations convenient only. Equation (44) becomes however this circumstance is not essential because, as we will E =+KPlt+>P()ωωF (45) see later, almost entire energy is due to the modes with high Equation (45) can be re-written using the virial theorem frequency propagating in the solid-like elastic regime anyway. Integrating (49), we find P = El and P ωω>=Et()ωω> F (here, P and E refer to l 2 t()F 2 −1 their average values) and by additionally noting that the total ⎛ N ⎞−1 ⎛ 2N ⎞ N N1 kinetic energy K is equal to the value of the kinetic energy of a ZT2 = ⎜∏∏ ωωli⎟ T ⎜ si⎟ (50) solid and can therefore be written, using the virial theorem, as ⎝i=1 ⎠ ⎝ωωti> 0 ⎠ the sum of kinetic terms related to longitudinal and transverse where N1 is the number of transverse modes with ω > ωF. waves: K =+EElt, giving 22 In the harmonic approximation, frequencies ωli and ωti are considered to be temperature-independent, in contrast to EEtt()ωω> F E =+El + (46) anharmonic case discussed in the next section. Then, equa- 22 tion (50) gives the energy E ==TZ2 d ln NT + NT. dT 1 Noting that Et can be represented as Ett= ωF), liquid energy reads in the Debye model, as is done in solids [2]. Here and below, the developed theory is at the same level of approximation Et()ωω< F E =+EElt()ωω>+F (47) as Debye theory of solids. The density of states of trans- 2 6N 2 verse modes is gt()ωω= 3 , where ωmt is Debye frequency The first two terms in (47) give the energy of propagat- ωmt ing phonon states in the liquid. The second term is the energy of transverse modes and we have taken into account that the of two transverse modes which decreases with temperature. number of transverse modes in the solid-like density of states This decrease includes both kinetic and potential parts, how- is 2N. ωmt can be somewhat different from the longitudinal ever the total kinetic energy of the system stays the same as in Debye frequency; for simplicity we assume ωmt ≈ ωD. Then, ωD 3 equation (44). The last term ensures that the decrease of the ⎛ ωF ⎞ N1 ==∫ gNt()ωωd2⎜⎟1 − . energy of transverse waves does not change the total kinetic ωF ⎝ ()ωD ⎠ energy, rather than points to the existence of low-frequency To calculate the last term in equation (47), we note that transverse waves (these are non-propagating in liquids). similarly to Et()ωω>=F1NT, Et()ωω< F can be calculated Either (46) or (47) can now be used to calculate the liquid to be Et()ωω<=F2NT, where N2 is the number of shear 3 ωF energy. Each term in equations (46) or (47) can be calculated modes with ω < ωF. Because N21=−2NN, N2 = 2N . as the phonon energy, E : ()ωD ph The total liquid energy is E =+NN+ N2 T according to 1 2 equation (47), giving finally [97()]: Eph = ∫ ET()ωω,dg()ω (48) ⎛ 3⎞ where g()ω is the phonon density of states. ⎛ ωF ⎞ E =−NT⎜3 ⎜ ⎟ ⎟ (51) Lets consider equation (47) and let Z2 be the partition function ⎝ ⎝ ωD ⎠ ⎠ associated with the first two terms in equation (47). Then, Z2 is: At low temperature where ττ D, or ωFD ω , equa- N ⎛ 1 ⎞ tion (51) gives c ==1dE 3, the harmonic solid result. At Z =−2eπω −N′ xp⎜ pq2 + 22⎟ddpq v N dT 2 ()∫ ⎜ ∑()i li i ⎟ ⎝ 2T i=1 ⎠ high temperature when ττ→ D and ωFD→ ω , equation (51) gives c = 2, consistent with the experimental result in fig- ⎛ 2N ⎞ v 1 2 22 ure 2. As the number of transverse modes with frequency ×−exp⎜ ∑ ()pq+ ωti ⎟ddpq (49) ∫ ⎜ 2T i i ⎟ above decreases with temperature, c decreases from ⎝ ωωti> F ⎠ ωF v

19 Rep. Prog. Phys. 79 (2016) 016502 Review about 3 to 2. A quantitative agreement in the entire tempera- in the Introduction. g(r) and U(r) featuring in equation (1) G∞ are not generally available apart from simple systems such ture range can be studied by using equation (15) or ωF = , η as Lennard Jones liquids. For simple liquids, g(r) and U(r) where is taken from the independent experiment. This way, – η can be determined from experiments or simulations and sub- E in equation (51) and cv have no free fitting parameters. The sequently used in equation (1). Unfortunately, neither g(r) agreement of equation (51) with the experimental cv of liquid nor U(r) are available for liquids with any larger degree of Hg is good at this level of approximation already [97]. complexity of structure or interactions. For example, many- In this picture, the decrease of cv with temperature is due body correlations [98, 99] and network effects can be strong to the evolution of collective modes in the liquid, namely the in familiar liquid systems such as olive oil, SiO2, Se, glyc- reduction of the number of transverse modes above the fre- erol, or even water [101], resulting in complicated structural 1 quency ωF = . We will discuss experimental data of c for sev- τ v correlation functions that cannot be reduced to the simple eral types of liquids in more detail below, including metallic, two- or even three-body correlations that are often used. As noble and molecular liquids, and will find that their cv similarly discussed in [6], approximations become difficult to con- decreases with temperature as equation (51) predicts. The same trol when the order of correlation functions already exceeds trend, the decrease of cv with temperature, has been experimen- three-body correlations. Similarly, it is challenging to extract tally found in complex liquids, including such systems as tolu- multiple correlation functions from the experiment. The ene, propane, ether, chloroform, benzene, methyl cyclohexane same problems exist for interatomic interactions, which can and cyclopentane, hexane, heptane, octane and so on [106]. be equally multibody and complex, and consequently not We have focused on calculating liquid energy and resulting amenable to determination in experiments or simulations. heat capacity that have contributions from collective modes On the other hand, ωF (τ) is available much more widely and diffusing atoms. We have not discussed liquid entropy as discussed above, enabling us to calculate and understand which includes the configurational entropy measuring the liquid cv readily. total phase space available to the system, the phase space sam- Next, expressing the liquid energy in terms of ωF in equa- pled by diffusive particle jumps. Unlike entropy, the energy tion (51) represents a more general description of liquids as is not related to exploring the phase space, and corresponds compared to equation (1). In equation (1), the energy strongly to the instantaneous state of the system (in the microcanoni- depends on interactions. It was for this reason that Landau and cal ensemble, or averaged over fluctuations in the canonical Lifshitz state that the liquid energy is strongly system-depend- ensemble). We will return to this point below when we discuss ent and therefore cannot be calculated in general form [2]. Let thermodynamic properties of viscous liquids. us now consider liquids with very different structural correla- We make two remarks related to using the Debye model. tions and interatomic interactions such as, for example, H2O, First, the Debye model is particularly relevant for disordered Hg, AsS, olive oil, and glycerol. As long as ωF of the above isotropic systems such as glasses [2], which are known to be liquids is the same at a certain temperature, equation (51) pre- nearly identical to liquids from the structural point of view dicts that their energy is the same (in molecular liquids, we are [34]. Furthermore, we have seen earlier that the dispersion referring to the inter-molecular energy as discussed below in curves in liquids are very similar to those in solids (includ- more detail). In this sense, expressing the liquid energy as a ing crystals, poly-crystals and glasses). Therefore, the Debye function of ωF only is a more general description because ωF model can be used in liquids to the same extent as in solids. is a uniformly common property for all liquids. One important consequence of this is that high-frequency Finally, equation (51), as well as its modifications below, modes in liquids make the largest contribution to the energy, are simple. This makes it fairly easy to understand and inter- as they do in solids including disordered solids. This is pret experimental data as discussed in the later section. re-iterated elsewhere in this paper. An objection could be raised that, although our approach Second, recall our earlier observation that ω gradually explains the experimental cv of liquids as discussed below, the increases from 0 to ω = ck around ωF with a square-root approach is based on ωF, the emergent property rather than on dependence (see equation (34) and discussion below). Writing the ostensibly lower-level data such as g(r) and U(r) in equa- ωD g d in the previous paragraph assumes a sharp lower tion (1). This brings us to an important question of what we ∫ω t()ωω F aim to achieve by a physical theory. According to one view, frequency cutoff at ωF, and is an approximation in this sense. The approximation is justified because it is the highest fre- ‘the point of any physical theory is to make statements about the outcomes of future experiments on the basis of results from quency modes above ωF that make the most contribution to the liquid energy, and because Debye density of states we employ the previous experiment’ [100]. This emphasizes relationships is already an approximation to the frequency spectrum, the between experimental properties. In this sense, equation (51) approximation that may be larger than the one involved in provides a relationship between liquid thermodynamic prop- substituting the square-root crossover with a sharper cutoff. erties such as energy and cv on one side and its dynamical and oscillatory properties such as ωF on the other.

Including anharmonicity and thermal expansion Comment on the phonon theory of liquid thermodynamics In calculating the energy E = TZ2 d ln , we have assumed We pause for the moment to make several comments about dT equation (51) and its relationship to our starting equation (1) that the phonon frequencies are temperature-independent.

20 Rep. Prog. Phys. 79 (2016) 016502 Review Generally, the phonon frequencies reduce with temperature. ⎛ ⎛ ωi ⎞⎞ This takes place at both constant pressure and constant vol- Fph =+ET0 ∑ ln⎜1e−−xp⎜⎟⎟ (57) i ⎝ ⎝ T ⎠⎠ ume. At constant volume, reduction of frequencies is related to inherent anharmonicity and increased vibration ampli- where E0 is the energy of zero-point vibrations. In calculat- dF tudes. If frequencies are temperature-dependent, applying ing the energy, E =−FTph , we assume dωi ≠ 0 as in the ph ph dT dT E = TZ2 d ln to equation (50) gives dT previous section, giving for the phonon energy N N 1 dω 2 1dωli 2 1dωti ω − T i E2 =−NT T ∑∑+−NT1 T (52) i dT i==1 ωli dT i 1 ωti dT Eph =+E0 ∑ (58) i exp1 ωi − where the derivatives are at constant volume. Equation (52) ()T gives the first two terms in equation (47). d Using ωαii=− ω as before gives Using Grüneisen approximation, it is possible to derive a d2T 1d useful approximate relation: ωαi =− , where α is the ⎛ αωT ⎞ i ω d2T v Eph =+E0 ⎜⎟1 + ∑ () ωi (59) ⎝ 2 ⎠ i exp1− coefficient of thermal expansion [102, 103]. Using this in (52) T gives () In this form, equation (59) can be used to calculate each of ⎛ αT ⎞ the three terms in (47). The energy of one longitudinal mode, E21=+()NNT⎜⎟1 + (53) ⎝ 2 ⎠ the first term in equation (47), can be calculated by substituting the sum in equation (59), , with Debye vibrational The last term in equation (47), Et()ωτ< 1/ , can be calculated ∑ 2 3N 2 1 αT density of states for longitudinal phonons, g()ωω= 3 , in the same way, giving NT 1 + , where N is the num- ωD 2 2 2 2 () where ωD is Debye frequency. Integrating from 0 to ωD gives ber of shear modes with ω < ωF calculated in the previous sec- x 3 ωD 3 zzd tion. Adding this term to equation (53) and using N and N ∑ = NTD , where Dx()= 3 ∫ is Debye 1 2 ()T x 0 exp1()z − from the previous section gives the anharmonic liquid energy: function [2]. The energy of two transverse modes with frequency ω > ωF, the second term in equation (47), can be ⎛ 3⎞ ⎛ αωT ⎞ ⎛ F ⎞ E =+NT⎜⎟1 ⎜3 − ⎜ ⎟ ⎟ (54) similarly calculated by substituting ∑ with density of states ⎝ 2 ⎠⎝ ⎝ ωD ⎠ ⎠ 6N g ωω= 2, where the normalization accounts for the num- () ω3 which reduces to (51) when α = 0. D ber of transverse modes of 2N. Integrating from ωF to ωD gives Equation (54) has been found to quantitatively describe 3 ωωDFωF cv of 5 commonly studied liquid metals in a wider tempera- ∑ =−22NTDNTD. Finally, Et()ωω< F ()TT()ωD () ture range where cv decreases from about 3 around the melt- in the last term in equation (47) is obtained by integrating point to 2 at high temperature [104]. The presence of the ing ∑ from 0 to ωF with the same density of states, giving αT 3 anharmonic term in equation (54), 1 + , explains why ω ω ( 2 ) ∑ = 2NT F D F . Putting all terms in equation (59) and ωD T experimental cv of liquids may exceed the Dulong-Petit value ()() then equation (47) gives finally the liquid energy cv = 3 close to the melting point [21, 22]. At low temperature when ττ D, equation (54) gives ⎛ 3 ⎞ ⎛ αωT ⎞ ⎛ DF⎞ ⎛ ω ⎞ ⎛ ωF ⎞ EE=+0 NT⎜⎟1 +−⎜3D⎜⎟⎜ ⎟ D⎜⎟⎟ ⎛ αT ⎞ ⎝ 2 ⎠ ⎝ T ⎠ ⎝ ωD ⎠ ⎝ T ⎠ E =+31NT⎜⎟ (55) ⎝ ⎠ ⎝ 2 ⎠ (60)

and cv is In general, E0 is temperature-dependent because it depends on ωF and therefore T. However, this becomes important at cv =+31 αT (56) () temperatures of several K only, whereas below we deal with Equation (56) is equally applicable to solids and viscous significantly higher temperatures where E0 and its derivative liquids where ττ D, and has been found consistent with sev- in (60) are small compared to the second temperature-depend- eral simulated crystalline and amorphous systems [102]. ent term. In the subsequent comparison of (60) with experi- We note that equations (55) and (56) don’t need to be mental cv, we therefore do not include E0. derived from equation (54), and also follow from considering In the high-temperature classical limit where ωD 1 and, the solid as a starting point where all three modes are present. T therefore, ωF 1 (ω < ω ), Debye functions become 1, and T FD (60) reduces to the energy of the classical liquid, equation (54). Including quantum effects For some of the liquids discussed in the next section, the If the temperature range includes low temperature where high-temperature classical approximation ωD 1 does not T ωD 1 does not hold, effects related to quantum excitations T hold in the temperature range considered [105]. In this case, become important. In this case, each term in equation (47) can quantum effects at those temperatures become significant, and be calculated using the phonon free energy [2] as equation (60) should be used to calculate liquid cv.

21 Rep. Prog. Phys. 79 (2016) 016502 Review We note that Debye model is not a good approximation in molecular and hydrogen-bonded systems where the frequency of intra-molecular vibrations considerably exceeds the rest of frequencies in the system (e.g. 3572 K in CO and 2260 K in O2). However, the intra-molecular modes are not excited in the temperature range of experimental cv (see figure 9). Therefore, the contribution of intra-molecular motion to cv is purely rotational, crot. The rotational motion is excited in the considered temperature range, and is classical, giving crot = R for linear molecules such as CO and O and c = 3R for mol- 2 rot 2 ecules with three rotation axes such as CH4. Consequently, cv for liquid CO shown in figure 9 corresponds to the heat capacity per molecule, with crot subtracted from the experimental data. In this case, N in equations (51), (54) or (60) refers to the number of molecules.

Phonon excitations at low temperature The number of excited phonon states increases with temperature. At low temperature, this results in the well-known Figure 9. Experimental c (black color) in metallic, noble v 3 and molecular liquids (kB = 1). Experimental cv are measured increase of cv: cv ∝ T . This increase can compete with the on isobars. Theoretical cv (red color) was calculated using decrease of cv to the progressive loss of transverse modes equation (60). The data are from [105]. The data for molecular and discussed above. In practice, all liquids solidify at low tem- noble liquids are taken at high pressure to increase the temperature perature and room pressure except helium. In liquid helium range where these systems exist in the liquid form [107]. We show the data in two graphs to avoid overlapping. under pressure, cv can first increase with temperature due to the phonon excitation effects. This is followed by the decrease of cv at higher temperature, similar to the behavior of classical Comparison with experimental data liquids in figure 9. As a result, cv can have a maximum [108]. The most straightforward comparison of the above theory to An interesting assertion can be made about the operation experiments is to calculate the energy using equations (51), of transverse modes in a hypothetical liquid in the limit of zero temperature: transverse modes do not contribute to liq- (54) or (60) and experimental ωF. This is often done by fitting uid s energy and specific heat in this limit 97[ ]. Indeed, let us ωF to function such as the VFT law, using it to calculate the ’ consider a liquid with a certain ωF and calculate the quantum energy and differentiating it to find cv and compare it to the experimental data. Equations (51), (54) or (60) involve no free energy of two transverse modes with frequency above ωF as fitting parameters, and contain parameters related to system t ωD ω t ET ωω>=F ∫ ω gt ωωd . ET ωω> F can be writ- ()ωF exp1− () () properties only. If ω is calculated from experimental viscos- T F ten as G∞ ity as ωF = , equation (51) contains G∞ and τD which enter η ωDF ωωggdωωω ωωd as the product G τ . Equation (54) contains parameters G τ t t() t() ∞ D ∞ D ET()ωω>=F ∫∫− (61) 00exp1 ω − exp1ω − and α. In equation (60), G∞ and τD feature separately. T T In the last few years, we have compared theoretical and 6N 2 experimental cv of over 20 different systems, including metal- Integrating (61) with Debye density of states gt()ωω= 3 ωD lic, noble, molecular and network liquids [97, 104, 105]. gives: We aimed to check our theoretical predictions in the widest 3 temperature range possible, and therefore used the data at t ⎛ ωωDF⎞ ⎛ ⎞ ⎛ ωF ⎞ ET()ωω>=F 22NTD⎜⎟− NT⎜ ⎟ D⎜⎟ (62) pressures exceeding the critical pressures from the National ⎝ T ⎠ ⎝ ωD ⎠ ⎝ T ⎠

Institute of Standards and Technology (NIST) database [107]. 4 In the low-temperature limit where Dx = π , the two As a result, many studied liquids are supercritical. In figure 9, () 5x3 t we show the comparison of theoretical and experimental data terms cancel exactly, giving ET()ωω>=F 0. The same result for several representative liquids. We have included three follows without relying on the Debye model and from observ- liquids in each class: metallic, noble and molecular liquids. ing that in the low-temperature limit, the upper integration We observe good agreement between experiments and limits in both terms in (61) can be extended to infinity due to theoretical predictions in a wide temperature range of about t fast convergence of integrals. Then, ET()ωω> F in (61) is the 50–1300 K in figure 9. The agreement supports the interpre- difference between two identical terms and is zero [97]. tation of the universal decrease of cv with temperature: the t Physically, the reason for ET()ωω>=F 0 is that only high- decrease is due to the reduction of the number of transverse frequency transverse modes exist in a liquid according to (11), modes propagating above frequency 1. but these are not excited at low temperature. τ

22 Rep. Prog. Phys. 79 (2016) 016502 Review We will re-visit this result in the later section discussing solid-like approaches to quantum liquids such as liquid helium.

Heat capacity of supercritical fluids

In the above discussion, cv decreases from about 3 at low temperature to 2 at high, corresponding to the complete loss of solid-like transverse modes. It is interesting to ask how cv changes on further temperature increase. On general grounds, one expects to find the gas-like valuec = 3 at high tempera- v 2 ture where the kinetic energy dominates. Figure 10. The Frenkel line in the supercritical region. Particle If the system is below the critical point (see figure 1), fur- dynamics includes both oscillatory and diffusive components ther temperature increase involves boiling and the first-order below the line, and is purely diffusive above the line. Below the 3 line, the system is able to support rigidity and transverse modes at transition, with cv discontinuously decreasing to in the gas 2 high frequency. Above the line, particle motion is purely diffusive, phase. The intervening phase transition excludes the state and the ability to support rigidity and transverse modes is lost at 3 all available frequencies. Crossing the Frenkel line from below of the liquid where cv can gradually change from 2 to and 2 corresponds to the transition between the rigid liquid to the where interesting physics operates. However, this becomes ‘ ’ ‘non-rigid’ gas-like fluid. possible above the critical point. This brings us to the interesting discussion of the supercritical state of matter. Qualitatively, the FL corresponds to ττ→ D (here, τD refers to the minimal period of transverse modes), implying that Frenkel line particle motion loses its oscillatory component. Quantitatively, Supercritical fluids started to be widely deployed in many the FL can be rigorously defined by pressure and temperature important industrial processes [109, 110] once their high dis- at which the minimum of the velocity autocorrelation function solving and extracting properties were appreciated. These (VAF) disappears [113]. Above the line defined in such a way, properties are unique to supercritical fluids and primarily velocities of a large number of particles stop changing their sign result from the combination of high density and high particle and particles lose the oscillatory component of motion. Above mobility. Theoretically, little was known about the supercriti- the line, VAF is monotonically decaying as in a gas [113]. cal state, apart from the general assertion that supercritical Another criterion for the FL which is important for our fluids can be thought of as high-density gases or high-temper- discussion of thermodynamic properties and which coincides ature fluids whose properties change smoothly with tempera- with the VAF criterion is cv = 2 [113]. Indeed, ττ= D cor- ture or pressure and without qualitative changes of properties. responds to the complete loss of two transverse modes at This assertion followed from the known absence of a phase all available frequencies (see equation (11)). The ability to transition above the critical point. support transverse waves is associated with solid-like rigid- We have recently proposed that this picture should be ity. Therefore, ττ= D corresponds to the crossover from the modified, and that a new line, the Frenkel line (FL), exists ‘rigid’ liquid to the ‘non-rigid’ gas-like fluid where no trans- above the critical point and separates two states with distinct verse modes exist [111–114, 124], corresponding to the quali- properties (see figure 10) [111–114]. The main idea of the FL tative change of the excitation spectrum. lies in considering how particle dynamics changes in response According to equation (51), ωFD= ω or ττ= D gives to pressure and temperature. Recall that particle dynamics in cv = 2. This corresponds to the qualitative change of the exci- the liquid can be separated into solid-like oscillatory and gas- tation spectrum in the liquid, the loss of transverse modes. like diffusive components. This separation applies equally to Therefore, we expect to find an interesting behavior of cv supercritical fluids as it does to subcritical liquids: increas- around cv = 2 and its crossover to a new regime. This is ing temperature reduces τ, and each particle spends less indeed the case as discussed in the next section. time oscillating and more time jumping; increasing pressure Due to the qualitative change of particle dynamics, the FL reverses this and results in the increase of time spent oscil- separates the states with different macroscopic properties, lating relative to jumping. Increasing temperature at constant consistent with experimental data [107]. This includes diffu- pressure (or decreasing pressure at constant temperature) sion constant, viscosity, thermal conductivity, speed of sound eventually results in the disappearance of the solid-like oscil- and other properties [111, 113, 114]. For example, the fast latory motion of particles; all that remains is the diffusive sound discussed earlier disappears above the FL due to the gas-like motion. This disappearance represents the qualitative loss of shear resistance at all available frequencies. Depending change in particle dynamics and gives the point on the FL in on the temperature and pressure path on the phase diagram, figure 10. Notably, the FL exists at arbitrarily high pressure the crossover of a particular property may not take place on and temperature, as does the melting line. the FL directly but close to it.

23 Rep. Prog. Phys. 79 (2016) 016502 Review We note a different proposal to define a line above the critical point, the Widom line. At the critical point, thermodynamic functions have divergent maxima. Above the critical point, these maxima broaden and persist in the limited range of pressure and temperature. This enables one to define lines of maxima of different properties such as heat capacity, thermal expansion, compressibility and so on. Close to the critical point, system properties can be expressed in terms of the correlation length, the maxima of which is the Widom line [115]. The physical significance of the Widom line was originally attributed to the effect of persisting critical fluctuations on system’s dynamical properties [115]. Following the detec- tion of PSD above the critical point [74, 116], the Widom line was proposed to separate two supercritical states where PSD does and does not operate [75] (see [111] for the discussion of extrapolating the line to high pressure and temperature where Figure 11. c (k = 1) as a function of temperature from the no maxima exist). The states with and without PSD were v B molecular dynamics simulation of the Lennard–Jones (LJ) liquid called ‘liquid-like’ and ‘gas-like’ because they resemble the using the data from [111]. Temperature is in LJ units. Density is presence and absence of PSD in subcritical liquids and gases. ρ = 1 in LJ units. The region of dynamical crossover at cv = 2 is The discussion of the effect of the Widom line on thermody- highlighted in red and by the arrow. namic, dynamical and transport properties followed (see, e.g. [117, 118]). distance a. Further increase of particle energy at higher tem- Persisting critical anomalies and fluctuations related to the perature increases the mean free path of particles L, the aver- Widom line certainly affect system properties close to the crit- age distance which the particles travel before colliding. L sets ical point. At the same time, the physical origin of the Widom the minimal wavelength of the remaining longitudinal mode, line and the FL is different, as evident from the above discus- λL: indeed, oscillation wavelength can only be larger than L. sion. A detailed discussion of this point is outside the scope of Therefore, the propagating longitudinal mode above the FL this review. Here, we include two brief remarks: (a) the FL is has the wavelengths satisfying not physically related to the critical point and critical fluctua- > L (63) tions and exists in systems where the boiling line and the criti- λ cal point are absent such as the soft-sphere system [113]; and We observe that the oscillations of the longitudinal mode (b) the persisting maxima of thermodynamic functions and the in (63) disappear with temperature starting with the highest frequency (smallest wavelength) above the FL, in interesting Widom line decay around (1.5–2)Tc and strongly depend on the property (e.g. heat capacity, compressibility and so on) contrast to the evolution of transverse modes in (11) where and on the path on the phase diagram (i.e. the location of the transverse modes disappear starting with the smallest fre- Widom line depends on whether the property is calculated quency. The difference of temperature evolution of collective along isobars, isotherms and so on) [119–122]. This is in con- modes below and above the FL is responsible for the crosso- trast to the FL which exists at arbitrarily high temperature and ver of cv at the FL discussed below. pressure and is property- and path-independent. The energy of the above longitudinal mode, El, can be calculated using equation (48) as

Heat capacity above the Frenkel line ωL (64) El = ∫ ET()ωω,dg()ω A confirmation of the above theoretical proposal that the spe- 0 cific heat undergoes a crossover around cv = 2 comes from 22π π molecular dynamics simulations in the supercritical state where ωL ==cc is the minimal frequency. λL L 3N [111, 123]. cv of the model Lennard–Jones liquid is shown in Taking g ωω= 2 as before and E ω, TT= in the clas- () ω3 () figure 11. We first observe a fairly sharp decrease of cv from D 3 3 ωL a about 3 to 2, similar to the previously discussed behavior in sical case gives El = NT , or NT . The total energy of ωD L () () 3 figure 9. This is followed by the flattening and slower decrease the system is the sum of the kinetic energy, NT, and potential at higher temperature. The crossover takes place at around 2 energy. Using the equipartition theorem, the potential energy cv = 2 as predicted. can be written as El. This gives the total energy of the non- Understanding the slower decrease of cv above the FL 2 involves the discussion of how the remaining longitudinal rigid gas-like fluid above the FL as mode evolves with temperature (recall that two transverse 3 1 ⎛ a ⎞3 modes disappear at the FL). When the FL is crossed from E =+NT NT⎜⎟ (65) below, particles lose the oscillatory motion around their 2 2 ⎝ L ⎠ quasi-equilibrium positions, and start undergoing purely dif- Just above the FL, La≈ . According to equation (65), this fusive jumps with distances comparable with the interatomic gives cv = 2, the result that also follows from the equation (51)

24 Rep. Prog. Phys. 79 (2016) 016502 Review describing the rigid liquid. When La at high temperature, equation (65) gives c = 3 as expected. v 2 The crossover of cv seen in figure 11 is therefore attributed to two different mechanisms governing the decrease of cv. Below the FL, cv decreases from the solid value of 3 to 2 due to the progressive disappearance of two transverse modes with frequency ω > ωF. Above the FL, cv decreases from 2 to the ideal-gas value of 3 due to the disappearance of the 2 remaining longitudinal mode starting with the shortest wavelength governed by L. Remaining long-wavelength longitudinal oscillations, sound, make only small contribution to the system energy and heat capacity. The softening of the phonon frequencies with temperature can be accounted for in the same way as in the case of subcritical fluids above (see equations (52) and (53)), giving [123]: Figure 12. cv as a function of the characteristic wavelengths λmax 3 3 1 ⎛ αTa⎞⎛ ⎞ (maximal transverse wavelength in the system) and λmin (minimal E =+NT NT⎜⎟1 + ⎜⎟ (66) longitudinal wavelength in the system) illustrating that most 2 2 ⎝ 2 ⎠⎝ L ⎠ important changes of thermodynamics of the disordered system take place when both wavelengths become comparable to the 3 The actual decrease of c between c = 2 and c = can fundamental length a. v v v 2 be calculated if temperature dependence of L is known. This dependence can be taken from the independent measurement and (65) and interpreting both of them in terms of wavelengths of the gas-like viscosity of the supercritical fluid: [46] (see figure 12). 1 The minimal frequency of transverse modes that a liquid η = ρuL¯ (67) 3 supports, ωF, corresponds to the maximal transverse wave- ωD τ length, λmax, λmax ==aa, where a is the interatomic where u¯ is the average velocity defined by temperature. ωFDτ Taking η from the experiment, calculating L using (67) and separation, a ≈ 1–2 Å. According to equation (51), cv remains using it in equation (65) or equation (66) enables us to calcu- close to its solid-state value of 3 in almost entire range of available wavelengths of transverse modes until ωF starts to late E and cv. This gives good agreement with the experimen- approach ωD, including in the viscous regime discussed below, tal cv for several supercritical systems, including noble and or when λmax starts to approach a. When λmax = a, cv becomes molecular fluids [123]. In these systems, cv slowly decreases with temperature as is seen in figure 11 in the high-tempera- cv = 2 according to equation (51) and undergoes a crossover ture range. to another regime given by equation (65). In this regime, the We note that in our discussion of liquid thermodynamics minimal wavelength of the longitudinal mode supported by throughout this paper, we assumed that the mode energy is T the system is λmin = L. According to equation (65), cv remains (in the classical case). This applies to harmonic waves. In close to the ideal gas value of 3 in almost entire range of the 2 weakly-anharmonic cases, the anharmonicity can be accounted wavelengths of the longitudinal mode until λmin approaches a. for in the Grüneisen approximation (see equation (54) and When λmax = a, cv becomes cv = 2, and matches its low- related discussion). If the anharmonicity is strong, the mode temperature value at the crossover as schematically shown in energy can substantially differ from T. This can include the figure 12. case of very high temperature or inherently anharmonic sys- Consistent with the above discussion, figure 12 shows that tems such as the hard-spheres system as an extreme example c remains constant at either 3 or 3 over many orders of magni- where heat capacity is equal to the ideal-gas value. v 2 tude of λ, including the regime of viscous liquids and glasses a discussed below, except when λ becomes close to 1 by order Heat capacity of liquids and system s fundamental a ’ of magnitude. length Figure 12 emphasizes a transparent physical point: modes with the smallest wavelengths comparable to interatomic sepa- The behavior of liquid cv in its entire range from the solid rations a contribute most to the energy and cv in the disordered value, c = 3, to the ideal-gas value, c = 3, can be unified and v v 2 systems (as they do in crystals) because they are most numer- generalized in terms of wavelengths. ous. Consequently, conditions λmax ≈ a for two transverse Lets consider cv in the rigid liquid state, equation (51) and modes and λmin ≈ a for one longitudinal mode, corresponding in the non-rigid gas-like fluid, (65). We do not consider anhar- to the disappearance of modes with wavelengths comparable monic effects related to phonon softening: these give small to a, give the largest changes of cv as is seen in figure 12. corrections (αT 1) to the energy in equations (54) and (66). The last result is tantamount to the following general asser- An interesting insight comes from combining equations (51) tion: the most important changes in thermodynamics of the

25 Rep. Prog. Phys. 79 (2016) 016502 Review

Figure 13. Evolution of transverse and longitudinal waves in disordered matter, from viscous liquids and glasses at low temperature to gases at high. The variation of colour from deep blue at the bottom to light red at the top corresponds to temperature increase. The figure shows that the transverse waves (left) start disappearing with temperature starting with long wavelength modes and completely disappear at the Frenkel line. The longitudinal waves (right) do not change up to the Frenkel line but start disappearing above the line starting with the shortest wavelength, with only long-wavelength longitudinal modes propagating at high temperature.

disordered system are governed by its fundamental length simulations where transverse modes are directly calculated a only. Because this length is not affected by disorder, this from the transverse current correlation functions [124]. assertion holds equally in ordered and disordered systems. Above the FL, the collective mode is the remaining longi- Interestingly, the above assertion does not follow from tudinal mode with the wavelength larger than L, and its energy the hydrodynamic approach to liquids. The hydrodynamic progressively decreases with temperature until it becomes approach works well at large wavelengths, but may not cor- close to the ideal gas. rectly describe effects at length scales comparable to a. Yet, The evolution of collective modes and related changes of as we have seen, this scale which plays an important role in liquid energy and heat capacity are intimately related to the governing system’s thermodynamic properties. change of microscopic dynamics of particles and the relative weights of diffusive and oscillatory components. We will Evolution of collective modes in liquids: summary return to this point below when we discuss the mixed state of liquid dynamics as contrasted to pure dynamical states of We can now summarize the above discussion of how collec- solids and gases. tive modes change in liquids with temperature. This is illustrated in figure 13. Viscous liquids Figure 13 illustrates that at low temperature, liquids have the same set of collective modes as in solids: one longitudi- In this section, we discuss how energy and heat capacity of nal mode and two transverse modes. In the viscous regime at viscous liquids can be understood on the basis of collective low enough temperature where ττ or ω ω , the liquid D FD modes. ‘Viscous’ or ‘highly-viscous’ liquids are loosely energy is almost entirely given by the vibrational energy due defined as liquids where to these modes, as discussed in the next section. On temperature increase, the number of transverse modes propagating ττ D (68) above the frequency ωF decreases. At the FL where particles More generally, viscous liquids are discussed as systems lose the oscillatory component of motion and start moving dif- that avoid crystallization and enter the glass transformation 2 3 fusively as in a gas, the two transverse modes disappear. This range. When τ exceeds the experimental time scale of 10 –10 s picture is consistent with the results of molecular dynamics and particle jumps stop operating during the observation

26 Rep. Prog. Phys. 79 (2016) 016502 Review

time (in the field of glass transition, particle jumps are often 1 2 m av EEdif ++dif ... + Edif referred to as ‘alpha-relaxation’), the system forms glass. This Edif = (71) 2 3 m defines glass transition temperature asτ ()Tg = 10 –10 s [34]. i Properties of viscous liquids have been widely discussed where Edif are instantaneous values of Edif featured in equa- av due to the interest in the problem of liquid-glass transition, the tion (70). Edif is E problem which consists of several unusual effects and includes tot Eav EE1 ++2 ... + Em persisting controversies (see, e.g. [34, 35, 41, 42, 51–55]). dif = dif dif dif (72) Understanding viscous liquids above Tg is thought to facilitate Etot Emtot ⋅ explaining effects involved in the actual liquid-glass transition i Each of the terms Edif in equation (72) is equal to τD, accord- at Tg and possibly effects below Tg [34, 35, 41, 51, 53–55]. Etot τ We find that in some respects, the glass transition problem ing to equation (70). There are m terms in the sum in equa- is more controversial that it needs to be. This is partly because tion (72). Therefore, the controversies emerged before good-quality experimental av E τ data became available. For example, the crossover from the dif = D (73) VFT to the Arrhenius (or nearly Arrhenius) behavior at low Etot τ temperature [57–59, 61, 62] removes the basis for discussing We therefore find that under the condition (68), the ratio of possible divergence and associated ideal glass transition at the the average energy of diffusion motion to the total energy is VFT temperature T0, as discussed above. negligibly small, as in the instantaneous case. Consequently, the energy of the liquid under the condition (68) is, to a very good approximation, given by the remaining vibrational part. Energy and heat capacity Similarly, the liquid constant-volume specific heat,c = 1dEl v,l N dT Perhaps unexpectedly, understanding basic thermodynamic is entirely vibrational in the regime (68): properties of viscous liquids such as energy and heat capac- vib ity is easier than of low-viscous liquids. It does not involve EEl = l expanding the energy into the oscillatory and diffusive parts vib (74) ccv,l = v,l as in equation (44) or integrating over the operating phonon states as in equations (48)–(50). The main results can be The vibrational energy and specific heat of liquids in the obtained on the basis of one parameter only, τD (or ωF ) using regime (68) is readily found. When regime (68) is operative, τ ωD vib vib a simple yet rigorous statistical-mechanical argument [125]. El to a very good approximation is El = 3NT (here and The jump probability for a particle is the ratio between the below, kB = 1). Indeed, a solid supports one longitudinal mode time spent diffusing and oscillating. The jump event lasts on and two transverse waves in the range 0 <<ω 1 . The abil- τ the order of Debye vibration period τ ≈ 0.1 ps. Recall that τ D 1 D ity of liquids to support shear modes with frequency ω > , is the time between two consecutive particle jumps, and there- τ fore is the time that the particle spends oscillating. Therefore, combined with ττ D in equation (68), implies that a viscous liquid supports most of the shear modes present in a solid. the jump probability is τD. In statistical equilibrium, this prob- τ Furthermore and importantly, it is only the high-frequency ability is equal to the ratio of diffusing atoms, Ndif, and the shear modes that make a significant contribution to the liquid total number of atoms, N. Then, at any given moment of time: vibrational energy, because the vibrational density of states is 2 NdifDτ approximately proportional to ω . Hence in the regime (68), = (69) vib N τ El = 3NT to a very good approximation, as in a solid. If E is the energy associated with diffusing particles, We now now consider equations (74) in harmonic and dif anharmonic cases. In the harmonic case, equations (74) give E ∝ N . Together with E ∝ N, equation (69) gives difdif tot the energy and specific heat of a liquid as 3NT and 3, respec- Edif τD tively, i.e. the same as in a harmonic solid: = (70) Etot τ h h EEl ==s 3NT Equation (70) implies that under condition (68), the con- (75) cch ==h 3 tribution of Edif to the total energy at any moment of time is vv,l ,s negligible. where s corresponds to the solid and h to the harmonic case. We note that equation (70) corresponds to the instantaneous In the anharmonic case, equations (74) are modified by the value of Edif which, from the physical point of view, is given by intrinsic anharmonicity related to softening of phonon fre- the smallest time scale of the system, τD. During time τD, the sys- quencies, and become equations (55) and (56) as discussed tem is not in equilibrium. The equilibrium state is reached when above. the observation time exceeds system relaxation time, τ. After Three pieces of evidence support the above picture. First, time τ, all particles in the system undergo jumps. Therefore, we experimental specific heat of liquid metals at low tempera- need to calculate Edif that is averaged over time τ. ture is close to 3, consistent with the above predictions Let us divide time τ into m time periods of duration τD [21, 22]. As experimental techniques advanced and gave τ av each, so that m = . Then, Edif, averaged over time τ, Edif, is access to high pressure and temperature, specific heats of τD

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many noble, molecular and network liquids were measured d ln Zdif in a wide range of parameters including in the supercritical dT 1 (76) d ln Z region [107]. Similarly to liquid metals, the experimental cv dT vib of these liquids was found to be close to 3 at low tempera- The liquid entropy, S =⋅d TZln Z , is: ture where equation (1) applies (see [105] for a compilation of dT ((vibdif)) the NIST and other data of cv for over 20 liquids of different d d types). S =+T ln ZZvibvln ib ++T ln ZZdifdln if (77) dT dT Second, condition ττ D becomes particularly pronounced The condition (76) implies that the third term in equa- in viscous liquids approaching liquid-glass transition where τD τ tion (77) is much smaller than the first one, and can be becomes as small as τD ≈ 10−15. Experiments have shown that τ neglected, giving in the highly viscous regime just above Tg, Cp measured at high frequency and representing the vibrational part of heat d S =+T ln ZZZvibvln ib + ln dif (78) capacity coincides with the total low-frequency heat capacity dT usually measured [126, 127], consistent with equation (74). Equation (78) implies that the smallness of Edif, expressed In the glass transformation range close to Tg, the two heat by equation (76), does not lead to the disappearance of all capacities start to differ due to non-equilibrium effects and entropy terms that depend on diffusion because the term ln Zdif freezing of configurational entropy, and coincide again below remains. This term is responsible for the excess entropy of Tg in the solid glass. liquid over the solid. On the other hand, the smallness of Edif Third, representing cv by its vibrational term in the highly does lead to the disappearance of terms depending on Zdif in viscous regime above T gives the experimentally observed dS g the specific heat. Indeed,c = T (here, S refers to entropy change of heat capacity in viscous liquids above T as com- v,l dT g per atom or molecule), and from equation (78), we find: pared to glasses below Tg. This is discussed in the next section. We recall that the only condition used to make the above d ⎛ d ⎞ d d c v,l =+T ⎜⎟T ln ZTvibvln ZTib + ln Zdif (79) assertions is equation (68). For practical purposes, this condi- dT ⎝ dT ⎠ dT dT tion is satisfied forττ 10 D. Perhaps not widely recognized, Using equation (76) once again, we observe that the third the condition ττ≈ 10 D holds even for low-viscous liquids such as liquid metals (Hg, Na, Rb and so on) and noble liq- term in equation (79) is small compared to the second term, uids such as Ar near their melting points, let alone for more and can be neglected, giving viscous liquids such as room-temperature olive or motor oil, d ⎛ d ⎞ d cv,l =+T ⎜⎟T ln ZTvibvln Z ib (80) honey and so on. dT ⎝ dT ⎠ dT Notably, the condition ττ 10 D corresponds to almost the entire range of τ at which liquids exist. This fact was not fully As a result, cv does not depend on Zdif, and is given by the appreciated in earlier theoretical work on liquids. Indeed, on vibrational term that depends on Zvib only. As expected, equa- lowering the temperature, τ increases from its smallest limit- tion (80) is consistent with cv in equation (75). 3 Physically, the inequality of liquid and solid entropies, ing value of ττ=≈D 0.1 ps to τ ≈ 10 s where, by definition, Sls≠ S , is related to the fact that the entropy measures the total the liquid forms glass at the glass transition temperature Tg. Here, τ changes by 16 orders of magnitude. Consequently, phase space available to the system, which is larger in the liq- τD uid due to the diffusional component present in equation (78). the condition 1, equation (68), or ττ10 D, applies in τ However, the diffusional component, ln Z , although large, is the range 103 10−12 s, spanning 15 orders of magnitude of . dif – τ slowly varying with temperature according to equation (76), This constitutes almost entire range of where liquids exist τ resulting in a small contribution to c (see equations (79) and as such. v (80)). On the other hand, the energy corresponds to the instantaneous state of the system (or averaged over τ), and is not related to exploring the phase space. Consequently, E = E , Entropy lvib yielding equation (76) and the smallness of diffusional contri- Although equation (73), combined with equation (68), implies bution to cv despite Sls≠ S . that the energy and cv of a liquid are entirely vibrational as in We note that the common thermodynamic description of a solid, this does not apply to entropy: the diffusional compo- entropy does not involve time: it is assumed that the obser- nent to entropy is substantial, and can not be neglected [125]. vation time is long enough for the total phase space to be Indeed, if Zvib and Zdif are the contributions to the partition explored. In a viscous liquid with large τ, this exploration sum from vibrations and diffusion, respectively, the total parti- is due to particle jumps, and is complete at long times t τ tion sum of the liquid is ZZ=⋅vibdZ if. Then, the liquid energy is only, at which point the system becomes ergodic. d d d E =⋅TZ2 ln ZT=+2 ln ZT2 ln ZE=+E If extrapolated below Tg, configurational entropy of viscous dTT((vibdif)) d vib dT difvib dif (here and below, the derivatives are taken at constant volume). liquids reaches zero at a finite temperature, constituting an av E Edif apparent problem known as the widely discussed Kauzmann Next, dif 1 from equation (73) also implies 1, where, Etot Evib paradox. More recently, issues involved in separating configu- for brevity, we dropped the subscript referring to the average. rational and vibrational entropy and interpreting experimen- Therefore, the smallness of diffusional energy, Edif 1, gives tal data became apparent, affecting the way the Kauzmann Evib

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particle jumps are not operative at Tg during the time of observation. Therefore, v at Tg is given by purely elastic response as in elastic solid. Then, we write vv+ PBel r = l 0 V l vg PB (81) = g 0 V g

0 0 where V l and V g are initial volumes of the liquid and the glass, vg is the elastic deformation of the glass and Bl and Bg are bulk moduli of the liquid and glass, respectively. Let ∆T be a small temperature interval that separates the liquid from the glass such that τ in the liquid, τl, is Figure 14. Heat capacity of Poly(a-methyl styrene) measured in ∆T 0 0 ττlg=+()TT∆ and 1. Then, V l ≈ V g. Similarly, the calorimetry experiments [128]. Glass transformation range operates Tg in the interval of about 260–270 K. difference between the elastic response of the liquid and the glass can be ignored for small ∆T, giving vvel ≈ g. Combining paradox is viewed and extent of the problem (see, e.g. [34, the two expressions in (81), we find: 128] and references therein). Bg Bl = (82) ε1 + 1

Liquid-glass transition vr where ε1 = is the ratio of relaxational and elastic response vel In the previous section, we have ascertained that in the viscous to pressure. regime ττ D, liquid energy and heat capacity are essentially The coefficients of thermal expansion of the liquid and the given by the vibrational terms. What happens to heat capacity glass, αl and αg, can be related in a similar way. Lets consider when temperature drops below the glass transition tempera- liquid relaxation in response to the increase of temperature by ture Tg and we are dealing with the solid glass, the non-equi- ∆T. We write librium system where τ exceeds observation time? 1 vvel + r Figure 14 gives a typical example of the change of the αl = l V 0 ∆T constant-pressure specific heat, cp, in the glass transformation l g 1 vg range around Tg. If cp and cp correspond to the specific heat αg = (83) V g ∆T above and below Tg on both sides of the glass transformation 0 cl p where vel and vr are volume changes due to solid-like elas- range, g = 1.1–1.8 for various liquids [54, 129]. The change cp tic and relaxational response as in equation (81) but now in of c at T is considered as the thermodynamic signature p g ‘ ’ response to temperature variation and vg is elastic response of of the liquid-glass transition, and serves to defineT in the g the glass. Combining the two expressions for αl and αg and calorimetry experiments. Tg measured as the temperature of 0 0 assuming V l = V g and vvel = g as before, we find the change of cp coincides with the temperature at which τ 2 3 reaches 10 –10 s and exceeds the observation time. ααl2=+()ε 1 g (84) Most researchers do not consider the change of c as a sig- p vr where ε2 = is the ratio of relaxational and elastic response nature of the phase transition. This is supported by the numer- vel ous data testifying that the structure of the viscous liquid to temperature. above Tg and the structure of glass are nearly identical. What Equations (82) and (84) describe the relationships between causes the change of cp at Tg? B and α in the liquid and the glass due to the presence of par- Recall that liquid response includes viscous response ticle jumps in the liquid above Tg and their absence in the glass related to diffusive jumps and solid-like response. When the at Tg, insofar as Tg is the temperature at which t < τ. Consistent viscous response stops at Tg during the experimental time with experimental observations, these equations predict that scale (from the definition ofT g) and only the elastic response liquids above Tg have larger α and smaller B as compared to remains, system’s bulk modulus B and thermal expansion below Tg. coefficient α change. This results in different cp above and We are now ready to calculate cp below and above Tg. below Tg [103]. In the previous section, we have seen that in the highly viscous Lets consider that pressure P is applied to a liquid. regime, cv is given by the vibrational component of motion According to the Maxwell-Frenkel viscoelastic picture, the only (see equation (80)). Hence, we use cv =+31()αT from change of liquid volume, v, is vv=+el vr, where vel and vr equation (56) which accounts for phonon softening due to are associated with solid-like elastic deformation and viscous inherent anharmonicity. Writing constant-pressure specific 2 relaxation process. Lets now defineT g as the temperature at heat cp as cpv=+cnTBα , where n is the number density, which τ exceeds the observation time t. This implies that we find cp above and below Tg as

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According to equation (86), Tg increases with the logarithm of the quench rate q. In particular, this increase is predicted to be faster than linear with ln q. This is consistent with experiments [37, 131] and the data in figure 15. We note that equation (86) predicts no divergence of Tg because the maximal physically possible quench rate is set by the minimal internal ∆T time, Debye vibration period τD, so that in equation (86) is τD always larger than q. Can the ‘glass transition line’ be identified on the phase diagram separating the combined oscillatory and diffusive particle motion above the line from the purely oscillatory motion observed below Tg during the experimental time scale? This would serve as the opposite to the Frenkel line which separates the combined oscillatory and diffusive particle motion below the line from the purely diffusive particle motion above the line at high temperature. As we have seen above, Tg depends on the observation time (or frequency), and so no well-defined glass Figure 15. Increase of Tg in Pd40Ni40P19Si1 glass with the quench transition line exists on the phase diagram because the low- rate q [131]. temperature state is a non-equilibrium liquid. The Frenkel line, on the other hand, separates two equilibrium states of matter. To summarize this section, we have seen that several

l 2 important experimental results of the glass transition, includ- cTp =+31()ααl +>nT l BTlg, T ing the heat capacity jump and dependence of T on the quench g g 2 (85) rate can be understood in the picture viewing the glass as the cTp =+31()ααg +

30 Rep. Prog. Phys. 79 (2016) 016502 Review and the high-pressure polymeric liquid. This is shown in the phase diagram in figure 16(b). As for Se, the line of liquid– liquid transformation is terminated at very high temperature only and above 2200 °C. The key to understanding these transitions lies in liquid dynamical properties. Indeed, if, as was often the case in the field, we consider a liquid as a structureless dense gas, no sharp phase transformations are possible. On the other hand, the oscillatory-diffusive picture of liquid dynamics based on τ offers a different insight. In the regime ττ D, particles perform many oscillations around fixed positions before jumping to the nearby quasi-equilibrium sites. Therefore, a well-defined short- and medium-range order exists during time τ. In this case, liquids not far above the melting point can support pressure-induced sharp or smeared structural changes, similarly to their solid analogues. Interestingly, the critical point of the liquid–gas transition in phosphorus is around 695 °C and 8.2 MPa. This means that the transition between the molecular and polymeric fluids takes place in the supercritical state. This may have come as a surprising finding in view of the perceived similarity of the supercritical state in terms of physical properties. Yet, as discussed above, well-defined short- and medium-range order exist in liquids above the critical point as long as the system is in the ‘rigid’-liquid state below the FL where ττ D. We can Figure 16. (a) Sharp changes in the structure factor in liquid therefore predict that liquid–liquid transitions in the supercrit- phosphorus in a narrow range of pressure and temperature. (b) ical state operate in the rigid-liquid below the FL but not in the Phase diagram showing the transition line between molecular and polymeric phosphorus. The data are from [149]. non-rigid gas-like fluid state above the line. Not surprisingly and similar to solids, liquid–liquid phase [146], P2O5 [148]), halogenides (e.g. AlCl3 [142], ZnCl2 [142], transitions are accompanied by the change of spectrum of col- AgI [143]), and chalcogenides (e.g. AsS [47], As2S3 [147], lective modes. Recently, the evidence for this has started to GeSe2 [144]). Pressure-induced transformations are accompa- come from experiments and modeling [152]. nied by structural changes in both short-range and intermedi- Understanding liquid structure and its response to pressure ate-range order as well as changes of all physical properties. and temperature will continue to benefit from the develop- Moreover, multiple pressure-induced phase transitions may ment of experimental techniques and in situ experiments in take place in one system: for example, AsS undergoes the trans- particular (see, e.g., [153, 154] for review). formation between the molecular and covalent liquid, followed by the transformation to the metallic phase [47]. The transfor- Quantum liquids: solid-like and gas-like approaches mations take place in the narrow pressure range and with large changes of structure and major properties such as viscosity. A quantum liquid is a liquid at temperature low enough Transformations in simple liquids can be both sharp and where the effects related to particle statistics, Bose–Einstein smeared. The analysis suggests that sharp transitions take or Fermi-Dirac, become operable. Quantum liquids is a large place in liquids whose parent crystals undergo phase transi- area of research (for review, see, e.g. [2, 3, 155–158]), largely tions with large changes of the short-range order structure and stimulated by superfluidity in liquid helium. Here, we point to bonding type [151]. gas-like and solid-like approaches to quantum liquids and to One of the first examples of sharp liquid–liquid transitions similarities and differences of these approaches to those used is the semiconductor-metal transformation in liquid Se [139]. in classical liquids discussed earlier. The transition is accompanied by the change of the short-range The solid-like approach to the thermodynamics of quan- order structure, volume and enthalpy jumps as well as by very tum liquids is due to Landau. Emphasizing strong interac- large jump of conductivity. Near the melting curve, the transitions in the liquid and rejecting earlier proposals which did tion occurs at 700 C and 4 GPa. At very high temperatures not, Landau asserted that the energy of a low-temperature this transition becomes smooth and finally almost disappears. quantum liquid, such as liquid helium at room pressure, is the Another clear example of the sharp liquid–liquid transition energy of the longitudinal phonon mode [2]. in a simple isotropic system is the transition in liquid phos- In this consideration, the quantum nature of the liquid phorus [138, 149]. In figure 16 we show sharp changes of the simplifies the understanding of its thermodynamics: Landau structure factor taking place in a narrow range of pressure and argued that any weakly perturbed state of the quantum sys- temperature. An abrupt and reversible structural transforma- tem is a set of elementary excitations, or quasi-particles. In tion takes place between the low-pressure molecular liquid the low-temperature quantum liquid, the quasi-particles are

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phonons and are the lowest energy states in the system. This 2 p + 1 pp′′ H =+aa Ua12 ++aaa gives the solid-like heat capacity of a quantum liquid equal ∑ p p ∑ pp12 pp′′pp21 (88) p 2m 2 12 to that in the quantum solid but with one longitudinal mode only [2]: where the first and second terms represent kinetic and potential + pp′′ energy, a , a are creation and annihilation operators and U 12 2 p p pp12 2π n 3 cv = T (87) is the matrix element of the pair interaction potential U(r). 15 u 3 () Without the second term, the ground state of the system is where n is the number density and u is the speed of sound. the BEC gas state. For weak interactions, the energy levels of Equation (87) is in agreement with the experimental heat the system can be calculated in the perturbation theory. As a capacity of liquid helium at room pressure. result, the diagonalised Hamiltonian reads [2]: Interestingly, Landau assumed that only one longitudinal HE pb+b mode contributes to the energy of a low-temperature quantum =+0 ∑ ε()p p liquid and did not consider high-frequency transverse modes 2 2 22 ⎛ p ⎞ predicted earlier by Frenkel. This has been consistent with the ε()pu=+p ⎜ ⎟ 2m absence of direct experimental evidence of transverse modes ⎝ ⎠ in liquid helium at room pressure. However, it is interesting to 4π2na u = (89) ask whether one should generally consider transverse modes in m2 a hypothetical low-temperature liquid. As we have seen earlier m where n is concentration, aU= 0 and U0 is the volume (see section ‘phonon excitations at low temperature’ above), 4π 2 transverse modes do not contribute to the liquid energy in the integral of the pair interaction potential. limit of zero temperature. Hence Landau’s assumption turned According to equation (89), the presence of interactions out to be correct. modifies the energy spectrum of the Bose gas and results in In addition to explaining the experimental heat capac- the emergence of the low-energy collective mode with the ity, the solid-like phonon picture of liquid helium explained propagation speed u. At small momenta, ε = up. superfluidity. Superfluidity emerges due to the impossibility This result is analogous to the gas-like approach to clas- to excite phonons in the liquid moving slower than the critical sical liquids where the weak interactions result in the low- velocity. In the original Landau theory, the critical velocity is frequency sound. What happens when interactions are strong the speed of sound. Considerably lower critical velocity found as in liquid helium and when the perturbation theory does not experimentally was later attributed to other effects such as apply? Here, we face the same problem of strong interactions energy-absorbing vortices. as in the classical case. The above low-temperature picture is discussed in the lin- Landau rejected the possibility of BEC in a strongly- ear dispersion regime, ε = cp. At higher temperatures, higher interacting system: in his view, the low-energy states of the phonon branches become excited, including the roton part of strongly-interacting system are collective modes rather than the spectrum. Interestingly, the roton part, originally thought single-particle states as in gases, the picture similar to quan- to be specific to helium, later discussed in the context of the tum solids where phonons are the lowest energy states and Bose–Einstein condensate (BEC) [155] and thought to be where BEC is irrelevant. In later developments, BEC was unusual in more recent discussions [158], is seen in many generalized for the case of strongly-interacting system on the classical high-temperature liquids (see, e.g. [76] and [70] and basis of macroscopic occupation of some one-particle state. figure 7). It was estimated that in low-temperature liquid helium, about In addition to the solid-like approach to liquid helium men- 10% of atoms are in the BEC state while the rest is in the tioned above, the hydrodynamic approach has been widely normal state (in this picture, the interactions ‘deplete’ BEC) used to discuss hydrodynamic effects (naturally) such as den- [156]. The BEC component is then related to the superfluid sity waves (first sound) and temperature or entropy waves component, and its weight changes with temperature. (second sound) and their velocities [3, 156]. Interestingly and It is probably fair to say that compared to well-studied similar to the classical liquids, two regimes of wave propa- effects of BEC in gases, operation of BEC in liquids is not gation and two sounds are distinguished depending on ω. understood in a consistent and detailed picture. Pines and Waves with ωτ < 1 are in the hydrodynamic regime, and are Nozieres remark [156] that a quantitative microscopic theory referred to as the first sound. Regimeω τq > 1 corresponds to of liquid helium is yet to emerge. Leggett comments on the the ‘quasi-particle’ sound, where τq is the lifetime of the quasi- challenge of obtaining direct experimental evidence of BEC particle excitation [156], and is analogous to the solid-like in liquid helium as compared to gases [157]. elastic modes in classical liquids discussed above. We now recall our starting picture of liquids where particle Interesting problems related to gas-like versus solid- motion includes two components: oscillatory and diffusive. approach emerge when the question of BEC in liquid helium Can quantum liquids be understood on the basis of these two is considered. As in the previous discussion, we can identify types of motion only, similarly to classical liquids? An inter- two approaches: gas-like and solid-like. The gas-like approach esting insight has come from path-integral simulations [159]: is due to Bogoliubov, and starts with the Hamiltonian describ- the emergence of macroscopic exchanges of diffusing atoms ing weakly perturbed states of the Bose gas: contributes to the λ-peak in the heat capacity, confirming the

32 Rep. Prog. Phys. 79 (2016) 016502 Review earlier Feynman result [160], and is related to momentum In solids, particle motion is purely oscillatory, correspond- condensation and superfluidity. ing to R = 0. Indeed, τ → ∞ in ideal crystals or becomes This picture enables one to adopt the solid-like approach astronomically large in familiar glasses such as SiO2 at room to quantum liquids (instead of the commonly discussed gas- temperature [103]. like approach): we approach the system from the solid state In gases, particle motion is purely diffusive. This corre- where strong interactions and resulting collective modes are sponds to R = 1, as is the case in the supercritical state above considered as a starting point, and introduce diffusive parti- the FL where the oscillatory component of particle motion is cle jumps as in the classical case. From the thermodynamic lost and where ττ≈ D. point of view, these jumps only modify the phonon spectrum We note that R = 1 at the FL above the critical point or in classical liquids. In quantum liquids, they additionally con- in subcritical liquids constitutes a microscopic and physi- tribute to the exchange energy because particle jumps enable cally transparent criterion of the difference between liquids the effect of quantum exchange [161]. and gases [112]. Indeed, existing common criteria include Can the exchange energy be related to exchange frequency distinctions such as that gas fills available volume but liquid ωF, as is the case for liquid energy in equations (51), (54) or does not, or that gas does not possess a cohesive state but liq- (60)? This would amount to a Frenkel reduction discussed ear- uid does. These criteria are either not microscopic, are tied lier but applied to the exchange energy. We think interesting to a particular pressure range or can not be implemented in insights may follow. We feel that generally developing closer practice [112]. On the other hand, asserting that the gas state ties between the areas and tools of classical and quantum liq- is characterized by purely diffusive dynamics of particles uids should result in new understanding. whereas the liquid state includes both diffusive and oscillatory components of particle motion gives a microscopic and physically transparent criterion. Mixed and pure dynamical states: liquids, solids, We therefore find that R = 0 and R = 1 give solids and gases gases as two limiting cases of dynamical properties. In this sense, gases and solids are pure states of matter in terms of The emphasis of our review has been on understanding experi- their dynamics. It is for this reason that they have been well mental and modeling data and on providing relationships understood theoretically. Liquids, on the other hand, are a between different physical properties. In addition to this rather mixed state in terms of their dynamics, the state that combines practical approach, we can revisit a more general question solid-like and gas-like motions. It is the mixed state which has alluded to in the Introduction: how are we to view and classify been the ultimate problem for the theory of liquids. liquids in terms of their proximity to gases or solids? Throughout On the basis of R-parameter, we see that liquids can only the history of liquid research, different ways of addressing this be viewed as solid-like or gas-like when R is either close to 0 question were discussed [1, 2, 4–8, 10, 12, 15, 16]. or 1. In all other cases, liquids are thermodynamically close On the basis of discussion in this paper, our answer is that to neither state. This becomes apparent from looking at the liquids do not need such a classification, or any other compart- experimental thermodynamic data such as in figure 9. This mentalizing for that matter. With their interesting and unique highlights our earlier point about the distinct mixed dynamical properties, liquids belong to a state of their own. Throughout state of liquids and associated rich physics. this review, we have seen that most important properties of Once the last assertion is appreciated, theorists become liquids and supercritical fluids can be consistently understood better informed about what approach to liquids is more appro- in the picture where we are compelled to view them as dis- priate. The best starting point for liquid theory is to make no tinct systems in the notably mixed dynamical state. Particles assumptions regarding the proximity of liquids to gases or sol- undergo both oscillatory motions and diffusive jumps, and ids and seek no extrapolations of the hydrodynamic regime to the relative weight of the two components of motion changes the solid-like regime and vice versa. Instead, the best starting with temperature. As discussed in the section ‘viscous liquids’ point is to consider the microscopic picture of liquid dynam- above, this relative weight is quantified by the ratio τD, which ics and its mixed character from the outset, and recognize that τ we now define as parameter R: the relative weights of diffusive and oscillatory components change with temperature. Depending on the property in ques- τDFω R == (90) tion, we can encounter several possibilities. τ ωD If we are concerned with long-time and low-energy We have seen that R enters the energy of both low-viscous observables only (t > τ or ωτ < 1), the relevant equations are liquids (see equations (51), (54) and (60)) and highly-viscous hydrodynamic [3]. Well-understood, these equations describe liquids including in the glass transformation range (see equa- hydrodynamic properties independently and separately from tions (69), (70) and (73)) and is implicitly present throughout the solid-like regime. The solid-like approach to liquids does our discussion. not apply to hydrodynamic effects. In liquids, R varies between 0 and 1, and defines the liq- If we are interested in thermodynamic properties such uid’s proximity to the solid or gas state. This enables us to as energy and heat capacity, it is the solid-like properties of delineate solids and gases as two limiting cases in terms of liquids that matter most because high-frequency modes con- dynamics and thermodynamics. tribute to the liquid energy almost entirely and propagate in

33 Rep. Prog. Phys. 79 (2016) 016502 Review

the solid-like regime ωτ > 1. In this case, we can focus on simulations was originally rationalized by the difficulty to the solid-like regime of liquid dynamics from the outset and construct liquid theory [162], with simulations playing the treat it separately and independently from the hydrodynamic role of testing the theory. With the first simulation of liq- regime. In this approach, we do not need to extrapolate the uids performed in 1957, the generated data exceeds what is hydrodynamic description into the solid-like regime as is feasible to review. For more recent examples, an interested done in generalized hydrodynamics and where extrapolation reader can consult liquid textbooks and review papers cited schemes may be an issue. throughout this review. A common issue faced by computer Each regime, hydrodynamic or solid-like, can be analyzed simulations is the same as in experiments: understanding and separately. There are also mixed cases where, for example, interpreting the data. With reliable interatomic potentials we observe solid-like high-frequency modes (ωτ > 1) at long existing today, it is not hard to calculate cv shown in figure 9, times (t > τ) because we are interested in their propagation but understanding the results requires a physical model. As length. Here, we can start with either hydrodynamic or elas- far as liquid heat capacity is concerned, it is fair to say that ticity equations and modify them appropriately. This gives the MD simulations have not resulted in understanding liquid cv same results as we have seen above. such as shown in figure 9. Once liquid thermodynamics is better understood, MD simulations will provide interesting microscopic insights and potentially uncover novel effects. Conclusions and outlook These can include the operation of collective modes in the Our important conclusion regarding the theoretical view of solid-like elastic regime and their evolution at conditions not liquids has already been made in the previous section. In this currently sampled by experiments including in the supercriti- review, we discussed how this view evolved and how differ- cal state. ent ideas proposed at very different times were developing. We feel that bringing concepts and tools from classical and With the recent evidence for high-frequency solid-like modes quantum liquids closer may result in new understanding, par- in liquids, it has now become possible to use the solid-like ticularly in the area of thermodynamics and operation of BEC approach to liquids and discuss their most important thermo- in real strongly-interacting liquids. Exploring the mixed state dynamic properties such as energy and heat capacity. We have of liquid dynamics and the separation of solid-like oscillatory reviewed how this can be done for liquids in different regimes: and gas-like diffusive particle dynamics in quantum liquids low-viscous subcritical liquids, high-temperature supercriti- may bring unexpected new insights. cal gas-like fluids, viscous liquids in the glass transformation range and systems at the liquid-glass transition. In each Acknowledgments case, we have noted limitations and caveats of this approach throughout this review. We are grateful to S Hosokawa and A Mokshin for discussions As alluded to in the Introduction, liquids have been viewed and providing data and to EPSRC for support. as inherently complicated systems lacking useful theoretical concepts such as a small parameter. 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